I'd say that both are more or less interchangeable - both are right - "I haven't got" may be a little more colloquial: "I don't have it" "I haven't got it" both mean the same thing. Or "I don't have any money" "I haven't got any money" are fine too. I think I'd use "I don't have" more frequently though, but just as a matter of preference.

"I haven't got" and "I don't have" are both correct and standard. They both mean the same thing, but there's a pretty clear reason why both exist. I use both, but maybe I use "I haven't got" a little more when I'm speaking naturally.

I think you meant, I am poor, I don't have any money.