It depends what you mean by powerful, cult, and ancient - but there's an obvious example in the West, which is Christianity. In about 400 years, the messianic, pacifistic offshoot of an oppressed minority religion grew from a tiny, persecuted group to being the state religion of Rome, and stamping out most rival religions. Christianity certainly powerfully changed the course of the empire. I'm not sure how much of a problem they were, exactly - early Christians got in trouble a lot because they refused to show respect to the Imperial cult and the deified former emperors as gods, plus various other parts of Christian doctrine e.g. pacifism, Jews already having loads of issues with Rome, etc. that clashed with the demands of the Roman state, but I'm not certain how much actual trouble this caused for the Empire before Christianity started getting traction among the Emperors and causing rifts between Christians and Pagans (or between, say, Arian Christians and non-Arians) - factional issue certainly cause bloodshed on some scale for centuries, but I don't know if you can really call Christianity that late a cult, given how big it got.

I'm calling Christianity here a cult in the sense of a small, intense religion, rather than in the sense of a Roman mystery cult or in the modern connotation of a creepy scam religion. It might not be what you're thinking of but I think it qualified.

What do you mean by powerful, cult, and ancient, by the way? Could help narrow this down and get a more specific answer.

If by a problem you mean a threat to the ruling power, then certainly. Here are three examples:

The Yellow Turbans, a religious group led by a charismatic leader who believed that his movement would overthrow the Han dynasty. They rose up when then Han government had been weakened and helped hasten its decline.

The Taiping Rebellion, led by Hong Xiuquan, who after failing the imperial exam several tikes claimed to have spoken with Jesus, who claimed to be Hong's older brother. The rebellion was very bloody and occupied a significant amount of territory in Qing China.

The Ikko-Ikki, a group of Japanese peasants led by Buddhist monks who holed up in the mountains outside Kyoto and bogged down Nobunaga's conquest of the region by resisting fiercely.

Sects were always a threat to authority, especially if that authority had some sort of religious component, like most Medieval Kingdoms which were based on the religious idea of the monarchs being rulers by the grace of god. Besides the Asian examples already listed (Yellow Turbans, Ikko-Ikki, etc.) the middle ages are filled with religious sects which seriously butted heads with authority.

One example: The Hussite Wars. The followers of Bohemian/Czech priest Jan Hus were trying to create their own church based on the teachings of John Wycliffe. Basically, they tried to create their own christian religion in which priests wouldn't have had quite as much powers, since lay-men were allowed to perform the holy communion as well. The crackdown was basically a fullblown war between the King of Bohemia and the Hussites while also involving not one but two Crusades against the Hussites.

You might be interested in Livy's account of the worship of Bacchus in Rome. In Livy's version, the followers of Bacchus, coming from Greece, recruited young Romans to their secret rites, which offered them "debaucheries of every kind". Not only did the cult's worship involve orgies and human sacrifice, but its members used bribery, forgery, poisoning and murder to gain more power in Rome. The cult, Livy writes, had followers and supporters all through Rome, including in the highest levels of society; when the Senate learned of its existence, they were horrified by the threat it posed to Roman morals and took brutal measures to wipe it out.

(I'm using 'cult', by the way, in the sense of a 'mystery cult': a formal organization, dedicated to the proper worship of some god or goddess, which involved secret rites and secret knowledge that only those who had undergone initiation and bound themselves to the cult could access. This was not considered a negative thing - the secrecy was for reasons of religious purity, not because the cult necessarily had nefarious goals - and it was quite normal for people to be initiated into one (or several) of these cults. In the modern world, Scientology would fit the definition of a mystery cult; so would the Freemasons. Most branches of modern Christianity, which generally have no secret knowledge and are happy to share Christian texts and doctrine with anyone who asks, represent the opposite.)

Livy's account is certainly over-dramatized and likely much exaggerated, but a contemporary copy of the decree of 186 BC, in which the Roman Senate decreed limitations on the worship of Bacchus, still exists. In that decree, the Senate broke up the existing cult organization (for example, banning groups of larger than five people from meeting in worship, and forbidding them to have any sort of formal organizational structure such as a leader or a treasury) and authorized the death penalty for those who violate the restrictions they placed on Bacchic worship. This sort of religious repression is not common in Roman history (for comparisons, you might look at the Roman government's relationship with the cults of Isis and Mithras, other foreign gods who came to Rome), so something was going on with that cult which worried the Roman elite enough to take drastic action against it.