Were they largely powerless or used to manage imperial administration? Or was there a relationship between the Senate and the Emperor that was respected?
I cannot speak about the senate beyond the seventh century, but I can provide an overview of what we know c.500-700. It is easy to dismiss the body as a useless institution given the more tangible powers of the emperor and the army, but we should not underestimate its more informal powers, nor how their influence shifted over time. Historically speaking, it would be incorrect to generalise its status as either powerless or powerful!
The weakness of the senate is seemingly particularly clear during the reign of Justinian, as he imposed a vast reform programme that reshaped the bureaucracy, all with the help of a small cabal of advisors, not through consultation with the wider senate. According to Procopius, an opponent of the regime c.550:
The Senate, however, sat only for pictorial effect, with no power to vote or do anything. It was assembled as a matter of form, to comply with the ancient law, and none of its members was permitted to utter a single word. The Emperor and his Consort took upon themselves the decisions of all matters in dispute, and their will of course prevailed. And if anybody thought his victory in such a case was insecure because it was illegal, he had only to give the Emperor more money, and a new law would immediately be passed revoking the former one. And if anybody else preferred the law that had been repealed, the ruler was quite willing to re-establish it in the same manner.
This is of course a particularly grim picture of the senate's status and has led to some historians arguing for the senate's anger against the emperor as a result. It is often pointed out, for example, that as Justinian had literally been born a peasant, he was looked down upon by some within the bureaucracy and the Constantinopolitan aristocracy. This has contributed to the view that senators were widely involved in the Nika Riots (or Revolt if we are inclined) in 532 against Justinian. Yet from the sources it is not so clear that it was a senatorial plot; indeed, there are mentions of loyal senators being sent out by Justinian to assess the situation. Some senators, including one Origenes, were caught up in the moment and decided to raise up an usurper, but it is untenable, I think, to see the senate as a persistently disloyal group. Like anyone else, senators had a wide range of interests and should not be reduced to one-issue stereotypes. Senators could be incredibly blue-blooded and of venerable ancestry, or were newly elevated provincial newcomers, the bulwark of a new aristocracy of service. Even if they did not directly hold power within the bureaucracy, senators were still wealthy landowners and so had a say in local government and, by extension, at Constantinople too.
This was not so clear during the reign of Justinian, who had centralising tendencies, but it becomes more obvious under his successor, Justin II. One of his laws, for instance, granted landowners the right to select provincial governors; a seemingly minor concession, but considering that these politically-connected landowners were frequently senators, it was a significant gesture. Peter Sarris, in his Economy and Society in the Age of Justinian (2006), used this to argue for a unending battle between the senatorial aristocracy/landowning class against the imperial government, with victories going to both sides depending on the situation, but it is again I think too simple an explanation. Even in the dark days of Justinian, when imperial tyranny (in senatorial eyes) would have been at its strongest, the emperor’s edicts were still being sent out and enforced – they can be seen as anti-senatorial legislation, but they were nevertheless enacted by men no less wealthy and no less connected as the supposed ‘senatorial opposition’. The clearest sign of Justinian’s awareness of the senate’s power is I think his eventual decision to return confiscated property to the families who were involved in the Nika Riots – when the immediate crisis was over, the emperor was not interested in making an enemy of the senate and sought reconciliation. His overall relationship was certainly uneasy at times, but that is the case for any ruler and their powerful subjects. By the end of his reign, the senate was still clearly a force to be reckoned with, since Justin II had to repeatedly court the body in order to make up for his lack of experience (and, some would say, legitimacy).
Beyond this, we also know that the senate continued to wield significant powers. The senate was for example a court of law, and Justinian issued an edict in 537 confirming that all senators were allowed to attend these sessions, an innovation that surely strengthened the senators’ powers. Procopius’ words quoted above might be seen as a counter-argument, but it is now recognised that the historian’s words were written for a particular moment, when the Justinianic regime was seemingly on the cusp of being overthrown, so it should be seen as a document written to defame all aspects of the emperor’s rule, rather than anything particularly representative of what really happened. We of course should not privilege Justinian's law-codes over other evidence either, but too frequently Procopius’ position is the one taken by historians, when instead we should acknowledge the value of other sources as well.
Another important role for the senate is their involvement in the making of emperors. The classic formula, that acclamations by the people, the army, and the senate were necessary for an emperor to be seen as legitimate is a flawed one, especially with the recent publication of Anthony Kaldellis’ The Byzantine Republic: People and Power in New Rome (2015), but the senate nonetheless played its part and was certainly an important body that any potential emperor had to consider, even if they did not need their approval in every case. Most acclamations were not very dramatic, so we tend to overlook the senators' role, but when tensions were high, it is clear that senators were in the thick of the action. They were for instance involved in the struggle to find a new emperor in 518, the Nika Riots (as discussed above), and in bringing down Emperor Phocas in 608-10. In between, they also advised the emperor on important occasions, the most dramatic being the decision to aid the deposed Persian shahanshah, Khusro II, in 590. Again, it is true that the emperor overruled the senate’s advice, which was to abandon the Persian ruler to his fate, but the senate’s views were mentioned in three separate sources (all well-informed about events in Constantinople), so it is clear that some sort of consultation did take place.
This fact becomes clearer still in the seventh century. When Heraclius begged for peace with the Persians in 615, he did not do so in his own name, but instead got the senate to write a letter to the now resurgent Khusro, implicitly acknowledging that he was a secondary figure in the Roman state - the Persians had never recognised him, so when attempting to make peace on such humiliating terms, it was very necessary to make this distinction. The fact that the senate was involved in such a delicate mission is yet another clear sign of their importance. In the inimitable words of Sarris from his Empires of Faith (2011):
In what is perhaps the single most humiliating document in all of Roman history, the senators of Constantinople the New Rome begged Khusro’s forgiveness, excoriating Phocas as the murderer of Maurice. The senators described themselves as ‘insignificant men’, Khusro as their ‘supreme Emperor’, and the ambassadors as his ‘slaves’. Begging that the senatorial delegation be received and returned in safety, they entreated Khusro to consider Heraclius as his teknon, or ‘boy’—a word signifying both child and slave. In short, the Senate was requesting a negotiated surrender, such as would leave in place a rump Roman state, centred on Constantinople, but subject to openly acknowledged Persian overlordship. The realm of the Caesars, empire of Christ, and heir to the fiery genius of Alexander, would henceforth be a supplicant and tributary at the feet of the Persian, Zoroastrian shah. Khusro was not interested: the ambassadors were executed and Chalcedon seized. Constantinople itself was now firmly in his sights.
The empire did recover from this low-point, but the following decades continued to see senatorial involvement in government. The best example is the strife before and after the accession of Constans II, the grandson of Heraclius, in 641. After his father had died in 641, the general Valentine intervened and made Constans the sole ruler by deposing his step-uncle. The new emperor was however only a boy and needed help in his rule. Valentine was naturally included in the new regime, but the senate was now brought into government as well, as can be seen in Constans’ speech to the senate:
My father Constantine, who begot me, reigned for a considerable time in the lifetime of his father [...] and, after his death, for a very short time; for the envy of his stepmother Martina both cut off his fair hopes and deprived him of life, and this on account of Heraclonas, her illicit offspring by Heraclius. Your godly decision rightly cast her out from the imperial dignity along with her child lest the Roman Empire appeared to be ruled in an unlawful manner, as your superlative magnificence well knew. Wherefore I call upon you to be my expert counsellors regarding the common good of our subjects.