"I have got a woman at home." or "I've got a woman at home." are both correct grammatically.
"I have a woman at home." is also correct.
The grammar of "I got a woman at home." is not really correct by the standards of English teachers and/or grammar books. However everybody understands it, and it's very common to hear people say it that way casually in everyday situations where the speakers know that their grammar is not being judged.
Also when you are relaxed and speaking fast, "I got ..." sounds a lot like "I've got ...", which is fine.