He is a drug addicted, abusive, cheater.
Your child deserves so much better, you hold the key to your childs future in the palm of your hand. It is your responsibility to do the right thing for the child. Children should never be raised in a home of addiction and abuse. They carry their childhood into adulthood, IMO a parent must set a good example, a child would rather be raised by one good parent than suffer at the hands of an addict. Children hear and see everything.
He told you the truth, believe him. Read Codependent No More, start attending Naranon
meetings, read others posts and keep posting.
I am sorry that you are having to face the truth, but it is what it is.