For example, Henry VIII had hundreds of people killed for treason, most famously some of his wives and ministers. Likewise, his daughter Mary also had hundreds of protestants burned at stake for heresy.
Were the monarchs able to accuse anybody of crimes and expect them to be executed? Or did they had trials? If so, how much could they influence them?
Ah, now here's a complex question. The simplest answer I can give is kinda?It's important to understand the Tudor justice system. By this point judges were solidified as a professional legal role drawn largely from the broader legal class that also served as barristers. They also had juries, formal evidence, rules of procedure for how a trial should go, common law precedent and a diverse system of courts (arguably somewhat overly complex and specialised). Contrary to popular perception, in many cases where the death penalty applied judges and especially juries gave a certain amount of leniency to offenders to avoid its overuse, and there were a significant number of royal pardons issues (see for example "Law and Morality in Seventeenth Century England" Cynthia Herrup, 1985, and "Mercy and Authority in the Tudor State", K. J. Kesselring 2003) . However, in criminal cases it was generally the presumption of the law that offenders were guilty else the authorities would not have brought them for prosecution, so often this hope relied purely on discretionary leniency and would often still involve some punitive sentence. Corruption was also a problem, albeit sometimes in the defendants' favour.And outside of royal pardons, this was not much of an option in royal treason trials. Unless the monarch commutes your sentences or spares you altogether, your sentence for treason is always death.
Let's look at an example with Walter Raleigh's trial in 1603. Interestingly, Richard L. Marcus's " The Tudor Treason Trials: Some Observations on the Emergence of Forensic Themes" (1984 ) actually argues treason trials were in some ways more modern due to greater use of lawyers, longer procedures, and the fact many of the defendants were more able to defend themselves given their higher class education with individuals like Raleigh. Nonetheless, it is also the case that these were cases the Crown particularly wanted to secure certain verdicts with. And as Marcus notes, it was not uncommon for pressure to be applied, both during and after trials to encourage "correct" verdicts. Marcus notes that it was a common notion, indeed the origin of juries as something closer to witnesses, that juries should evaluate cases based on their own pre-existing knowledge and this should be a qualification for juries. So in a case like Raleigh's, he was majorly threatened by his own chronic unpopularity. Walter Raleigh, despite his modern reputation, was widely seen at the time as a lewd, cynical upstart, a dishonourable man, perhaps even atheist. So this idea of an informed jury very much worked against him because it meant the jurors were people who knew all the scurrilous gossip against him.
However, Marcus notes the verdict was greeted with sympathy despite Raleigh reputation. It was seen as very political and dubious. So people were capable of questioning a verdict, quite publicly even, it was not unheard of. It's also worth noting that the royal pardon exception applied. While both Raleigh and Cobham were sentenced as traitors, the new King James intervened, rather theatrically at the last minute when Raleigh was to be executed, to have Cobham and Raleigh merely imprisoned. Indeed, while in the tower Raleigh even had a writing desk and was in contact with James's heir, Prince Henry Frederick, an extraordinary degree of latitude for a man technically sentenced to death. So while James distrust of Raleigh and Cobham meant he and his courtiers pushed for them to be punished, and used this to remove them from court, the weak evidence and James' desire to appear merciful before his new subjects meant they settled for imprisonment. So Tudor monarchs were certainly not free from law and public opinion going against them. As Marcus notes some state treason trials actually resulted in a not guilty verdict. It was not possible, à la Joffrey in GOT, to summarily and without procedure have someone punished or killed. There were rules on what could be punished with what and how it should be. Even when justice was not not always justice, there was at least the pretence of it.