The short and unsatisfying answer is: it depends. It depends on the exact time period, and it depends on the specific location to an extent, in addition to whether we're talking about Buddhism or Daoism. Both had some sway on the governments, but Buddhism had a period of more famous scandal with regards court/bureaucratic influence.

For most of Chinese history Daoism is by far the more prominent player in the political and social landscape. This is mostly because in comparison to Buddhism (A) it was homegrown and (B) though Buddhism slightly predates/is contemporary with the origins of "formal" Daoism, Daoism exists as a much more complete and accessible philosophy for several centuries before Buddhism in China can claim the same status. I say "philosophy" rather than "religion" here because one could follow Daoist ideas, Confucian ideas, and local folk religious practices all perfectly compatibly without any sense of contradiction (and later you can throw Buddhism into the mix too). Really, from the Han to the Song dynasties, the philosophy/religion mix of Daoism is used by several of the dynasties and rulers to justify their rule and to take guiding principles from. This sort of hits the peak during the Song (with a revival in the Ming) whereby many emperors are prominent in both the promotion of Daoism and the support of the organized Daoist practitioners (who themselves were also interacting more with "non ordained" Daoist folk practitioners). These emperors often willingly sought council from the formal Daoist ministers and their opinions played a role in guiding ruling doctrines. The Tang dynasty actually directly claimed relation to the legendary founder of Daoism, Lao Zi, and thus the court was also intimately involved with Daoist ministers and Daoist ideas. Buddhism was also heavily influential at this time however, ironically enough.

During the 3 Kingdoms Period and the period of the Sui Dynasty (and much of the Tang), China was in a near-constant state of devastating warfare versus Chinese states, rebellions, and foreign powers. It's during this period that much of the populace sought guidance from a relatively new-to-China philosophy that was making the rounds: Buddhism. Buddhist texts in China at that time however mostly fell either in the categories of non existent, rare, and poorly translated. Many of the terms were commonly not direct translations from the original Sanskrit, but instead translated as analogous to Daoist and Confucian ideas, often regardless of whether or not they were truly analogous. During the Tang dynasty, a Buddhist monk named Xuanzang actually traveled to northern India for the purposes of collecting and notating many of the original Buddhist texts that existed and subsequently translating them into Chinese (this is what the later famous novel Journey to the West is based upon). During this time and especially following, Buddhism exploded in popularity in China. Buddhist monasteries rapidly popped up across the Tang empire and even the Tang ruling family themselves, those descendants of Lao Zi, often became devout and vocal Buddhists. It was simultaneously both the high point and the end of old Chinese Buddhism. The reasons for this were both economic and political, and were greatly intertwined with the greater state bureaucracy.

There are two major schemes that can be pointed to where the state used Buddhism for influence, and Buddhism used the state for influence. Buddhist monasteries of this time were tax free, much like religious institutions often are around the world today. Buddhist monasteries also were sites where agricultural and artisanal production occurred and often with the assistance of slave labor. Lords of large areas of land were known to "donate" a significant portion of this land to a new Buddhist monastery with the understanding that they would then receive kickbacks from the sale of products produced on monastic lands which generated great profits. Essentially this was a tax loophole. The owners of the monastery would also often use this arrangement to convince local lords and bureaucrats to give them even greater privileges and power, keeping the wealth flowing. The second scheme was devastating to the image of the imperial family. Though women didn't hold political office, women in the imperial court did have some semblance of power and ability to perform minor occasional functions. Several women of the court took it upon themselves to start issuing fraudulent licenses for Buddhist monastic practice, sidestepping most of the bureaucratic process and receiving large bribes in exchange. Large, opulent palaces were actually built by imperial family members with these funds as at least partially responsible. As a result of these things, and arguments that this was a foreign religion influencing what should be a government guided by Chinese principles, austerity measures and sweeping repressions were instituted later in the Tang, effectively all but totally ending Buddhism's influence in China for some time. Later forms of Buddhism would come up in prominence in subsequent dynasties, notably Chan Buddhism, but often was incorporated alongside or in a minor accompanying role to Confuician and Daoist ideologies, as far as state implementation and influence.

As /u/Jasfss has said, it depends. Within the bounds of my area, the Qing Dynasty (1636-1912), this depends in large part – but not entirely – upon whether you think Confucianism constitutes a 'religion'. By the time of the Qing, the dominant mode of Confucian thought was what we now term 'Neo-Confucianism', first espoused in the mid-Tang Dynasty and officially adopted under the Song and Ming. Neo-Confucianism eschewed more overtly personal spirituality, but nevertheless retained strong ritual and cosmological components. In line with the general Inner Asian approach, I've opted to view Confucianism as a religion for the purposes of statecraft, as this offers a useful way of viewing it in relation to Buddhism, Daoism and Tengri Shamanism in the Qing court.

It is important to note that the Qing Dynasty was never totally 'sinicised' – that is, made Chinese. Indeed, it can be questioned whether it was even slightly 'sinicised', or merely created a Chinese face for itself. I'm not going to dwell on this issue too much, but going forward it will be necessary to explain a key feature of the Qing system, which is its fundamental duality. On the one hand, there was a Chinese, Confucian aspect, one which laid out a circumscribed, collegial emperorship, and which emphasised rigidity, inflexibility and the superiority of the Chinese way of doing things; and on the other hand was an Inner Asian aspect, in which the monarch was paramount by divine fiat, but where there was also much more flexibility in the way that various peoples were handled, ultimately with the aim of being as inclusive as possible. One framework for viewing the first two centuries of the Qing – indeed, potentially even the entire dynasty – would be to see the central conflict of Qing history as being between these two partly compatible and also partly contradictory aspects, and the challenge of balancing both in order to ensure the integrity of the empire. Emperors had to present themselves to the Chinese as legitimated by the conventions of Neo-Confucianism in order to retain the manpower and resources that China provided, but also not acculturate to the extent of losing their Inner Asian ideological integrity, by which their rule over regions outside of China would be legitimated. Qing emperors essentially adopted a series of personas, each of which appealed to a distinct people or set of peoples, in order to maintain this balance.

It is with these personas that I would like to start. Virtually all of the personas held by a Qing emperor were, fundamentally, of some level of religious significance. In the Confucian sphere, he was Tianzi – Son of Heaven. This did not necessarily mean that he was some divinely-appointed, incontrovertible autocrat, indeed far from it. The emperor was a sage-king (shengwang) and his right to rule was not the result of possessing Heaven's favour, because in the Neo-Confucian worldview, Heaven was dispassionate and showed no such special favour. Rather, the Son of Heaven was such because he understood the will of Heaven and thus could serve as an intermediary between Heaven and Earth. This sagely quality was not unique to the emperor, however, and wise counsel was a key part of the necessary components of Confucian rule, for the emperor was not an infallible person, and his ministers thus exercised a degree of discretion in questioning his authority, even if they had little choice in carrying out his will once decided.

But in the Buddhist sphere, he was Chakravartin – the 'Wheel-Turner'. Rather than the incidental intermediary between a dispassionate Heaven and the unenlightened Earth, the Chakravartin was basically the avatar of Heaven's will, the hub around which the spokes of history turned. The Buddhist monarch was much more, well, monarchical, and by virtue of this sort of direct divine legitimation. Some notions of the Chakravartin had been part of the Chinese imperial persona, along with the Confucian Son of Heaven, since the Tang, but what made this role particularly stand out under the Qing was that their rule over Tibet and Mongolia (where most of the population were Tibetan Buddhist) meant that the mobilisation of the purer form of the Buddhist persona was especially pertinent. While the title of khagan of the Mongols had not originally been one with overt divine legitimation, the adoption of Tibetan Buddhism by the Mongols had led to the assumption of rulership coming to rely at least in part on a degree of acknowledgement by the Tibetan clergy. The most notable instance of this would be the Zunghar ruler Galdan, whose title of Boshoghtu Khan was specifically bestowed upon him by the Dalai Lama in 1679, despite his not being a descendant of Genghis Khan, which had traditionally been a requirement for assuming a Mongol khanship (his father, Batur, had been unable to become Khan of the Oirat due to not having Chinggisid blood). In turn, the Qing adoption of the khaganate continued these trappings.

Of course, he also held a special relationship with his Manchu warrior-administrator caste, upon whom his rule rested, and it was this set of relationships that was probably the least overtly 'religious'. While the Manchus had a term for 'Son of Heaven', that being abkai jui, the term that Manchus more typically used for the emperor was ejen, 'lord', and often more specifically enduringge ejen, 'sacred lord'. His position was to be that of the supreme head of what in the old Jurchen hierarchy had been a series of superior-inferior relationships expressed in the language of master and slave. Now, this had been formalised in the military hierarchy of the Banner system, but with the added point that all Manchus were the direct slaves (aha) of the sacred lord, for 'ejen' has another meaning – 'master'. Not only that, but during the Yongzheng reign, the title of ejen became reserved exclusively for the emperor. While there was not necessarily what might recognisably be termed 'religion' involved, for the Manchus these hierarchical relationships nevertheless carried a strong sense of sanctity, and the three emperors of the High Qing – Kangxi, Yongzheng and Qianlong – all displayed an interest, to a greater or lesser extent, in ensuring the proper delineation of relationships, both within and without the Banner system. But in addition to these human relationships was an animist spiritual tradition, whereby the Jurchen-Manchus, whose lifestyle revolved around hunting in addition to farming, defined themselves heavily by their relationship to nature. The symbolism of Manchu identity was permeated with the assignment of spiritual significance to elements of the natural world, particularly mountains and rivers. Such features, particularly the Sungari River and Changbai Mountains, would be the defining characteristics of the Manchu homeland, for there was, technically, no word for 'Manchuria' in Qing-era Manchu.

In short, then, being a Qing emperor meant the simultaneous wielding of multiple essentially sacral kingships. But holding these personas was not just a matter of laying claim to them. A huge amount of effort went into maintaining the legitimacy of holding these various positions through a vast amount of performative activity. The most obvious examples to look at would be religious ritual. The Qing palaces in Beijing, Chengde (Ma. Erdemu Be Aliha) and Mukden (Ch. Shenyang) were not merely residences, but also vast ritual complexes, kitted out with altars and cult sites through which the emperor conducted the rituals and ceremonies that confirmed the legitimacy of his various personas. Among the key functions were sacrifices to Heaven and Earth, conducted at the Temple of Heaven in the Forbidden City, and the imperial Manchu shamanic rites, which would be the first set of ceremonies performed each year, taking place on (lunar) New Year's Day. Mukden, the administrative capital and cultural centre of Manchuria, included a sanctuary of the cult of the Buddhist war god Mahākāla, strongly associated with the legitimation of the khagans allegedly dating back to the days of Khubilai Khan, and all new emperors would be inducted into this cult – and sometimes others – through tantric rituals, whereby the consciousness of great (usually martial) rulers of the past, such as Tang Taizong and Khubilai Khagan, would be symbolically transferred.

Aside from walking the walk, there was also plenty of talking the talk, which was something the Qing were particularly adept at. The aforementioned trio of High Qing emperors all engaged to some extent in maintaining a Confucian image through the issuing of texts assuring the Chinese populace of their right to rule as Confucian monarchs. The Kangxi Emperor's Sacred Edict of 1670 consisted of sixteen brief maxims, ostensibly aimed to promulgate correct behaviour, but also more subtly asserting that the emperor – only the second Manchu to rule China – had himself taken to heart the lessons of Confucianism. These maxims included words to the effect of 'maintain filial piety,' 'avoid heterodox religions,' and 'pay your taxes.' The Yongzheng Emperor, in response to the Zeng Jing treason case of 1727, issued the Discourse on Righteousness to Dispel Confusion, which asserted that attacks on the Manchus for being 'barbarians' were untrue – the Manchus had acculturated to Chinese, and most importantly Neo-Confucian, ways. The Qianlong Emperor disagreed with this idea, and in the Researches on Manchu Origins essentially asserted that there was no need for the Manchus to claim acculturation, as they were already legitimate exactly how they were before the conquest – hence their being able to carry said conquest out. What exactly about those origins made them legitimate is left somewhat vague, but it is interesting that Qianlong cited Mencius' assertions that the second sage-king Shun and the first Zhou king Wen were both barbarians as his main precedent for such a claim.