Both sentences work for me.
Student or Learner
I did some exercises on the use of 'wish' and according to the answer key it should be:
I like my new boss but I wish she would give me some more responsibility.
I think that it's possible to say:
I like my new boss but I wish she gave me some more responsibility.
Is it correct to use the past simple?
My above deleted message was "'gave' is the past subjunctive which means she doesn't give, is it correct?"
I'd say it's OK. It's the past subjunctive.
I should clarify a point about this. Recently, I think I said that a sentence like 1) "I wish she came to the party" was wrong. This might appear to be a contradiction, if 2) "I wish she gave me more responsibility" is correct.
The difference is that 1) refers to a one-time present situation which is counterfactual, and 2) refers to an ongoing present and future (or timeless) situation which is still possible. For 1) you need, "I wish she had come..." or "I wish she would come..." depending on whether the party is over yet.
I realise that other people use the subjunctive differently or not at all, so it's just my opinion.