Как правильно He hasn't got a cat или He haven't got a cat

Ответ: He hasn't got a cat.

Пояснение: глагол to have спрягается в Present Simple следующим образом:

I have, haven't

You have, haven't

He, she, it has, hasn't

We have, haven't

You have, haven't

They have, haven't

have got переводится "иметь", например: I have got a computer. You have got a smartphone.