Why is this correct:

I hope I haven't done that

But this is incorrect

I hope I hadn't done that.

And when using wish

I wish I hadn't done that (correct)

I wish I haven't done that (incorrect)

Vedi una traduzione

"I hope I hadn't done that" is wrong because you're jumping between tenses. "I hope" is in the present tense, but "hadn't done" is past perfect, so you're talking about a past event from the perspective of the past. If I say "I had done the chores before she walked in", I'm talking about an event in the past that happened before a reference point that is also in the past, but "I hope" is a reference point in the present so it doesn't make sense to say "I hope I hadn't done the chores badly", so you should instead say "I hope I haven't done the chores badly" as you're talking about a past event from a reference point in the present.

"I wish I hadn't done that" is talking about an unreal past event and so you're saying that you DID do the thing, but you regret doing it. You use the past perfect to talk about unreal situations, the same way you say "if I hadn't done that, I would be happy" rather than "if I haven't done that, I would be happy".