Plot hole: Beckett is framed for the murder of Vulcan Simmons by someone (likely connected to Senator Bracken) who has access to her backup weapon locked in a safe in her apartment. It's never really proven that she's innocent; the murder weapon is never recovered, and the real murderer is never brought to justice. And though she's still technically a murder suspect, Beckett is allowed - after finding the evidence for which she's been searching - to act in her law enforcement capacity and arrest Senator Bracken.
Suggested correction: She had evidence, (the tape) that he was a murderer and probably explained before the arrest. That should be enough to prove her innocence, and to also prove him guilty.
The tape is proof of Bracken's guilt, not of Beckett's innocence. And as we've seen, Bracken is too careful to get caught; he wouldn't clear her name just to help her, or without something in it for himself. People would start asking questions why, and the truth of his criminal wrongdoings would be exposed. And though it was later retconned in S8 that Mr. Smith helped clear her, it took two years for the writers to address it, so as a standalone episode, it's still a gaping hole in the story.
With proof on Bracken, that could also prove Beckett's innocence. It's two birds with one stone. Nobody believed her anyway and she was accused of murder, but once the news on Bracken gets out, nobody has any reason not to take her more seriously.
Continuity mistake: In the flashback of Beckett and Montgomery, when Montgomery finds her in the archive room studying her mother's murder case, Beckett says that she was unable to find any leads worth following. But in season three, when Montgomery explains to Castle that that encounter was when he first met Kate, he says "she found some things that didn't add up", suggesting that she did find a clue worth investigating.
Plot hole: On the recording that Captain Montgomery left for Beckett, the conversation is exclusively between Montgomery and Bracken. But as established in "Knockout", Montgomery was only a rookie during the Bob Armen incident; as two veterans who are more than capable of speaking up for themselves, it seems unlikely that Raglan and McCallister would choose a newbie as their voice, particularly to the assistant D.A who's blackmailing them.
Suggested correction: Maybe they thought that fewer things could go wrong with an "expert" of some sorts. Rookies tend to learn along the way, and do anything possible when starting out.