Welcome to the fandom!) And I think he does. Will and Hannibal have an interesting conversation about regrets in E9 of S2.
Will: Do you have any regrets?
Hannibal: With every choice lies the possibility of regret. However, if I choose not to do something, it’s usually for a good reason.
The “usually” in Hannibal’s reply implies that while he doesn’t have regrets often, sometimes he does make mistakes that he wishes hadn’t occurred. More than that, here’s the part that comes next.
Will: I’m riddled with regrets.
Hannibal: A life without regret would be no life at all.
We know Hannibal strives to live a full life; he knows its beauty and its pain, like he admitted, so he knows regrets, too.
In E7 of S3, Hannibal is writing formulas to turn back time when sitting next to Will’s bed. If he wants to go back, then there is something he wishes he could have done differently. Maybe this time, knowing what he does, he wouldn’t have broken the tea cup. His confusion about it is emphasized in E3 of S3:
Bedelia: Forgiveness is too great and difficult for one person. It requires two: betrayer and betrayed. Which one are you?
Hannibal: I’m vague on those details.
If he was asked this question a few months ago, he would have called himself the betrayed, but now that he had time to think, he’s no longer certain.
So yes, I’d say Hannibal regrets his mistakes.