So you're saying that he cheated on his ex wife with this lady, he is now with you and despite you explaining to him that you are not happy with his contact with this woman, he goes behind your back and is in contact with her still and blaming his actions on your "mental" state?
If I was in your position I would have left this jerk ages ago, his lack of consideration for your feelings and then trying to blame you for his actions shows that he really doesn't care about you.
Was about to add this and noticed your additional details and to be honest, you should leave the jerk for her sake. My Mum stayed with my Dad "because of the kids" and we suffered for it. We would have been better off, physically and emotionally, if she had off left him and let us have visitation contact as and when.
While you are letting him do this to you, your daughter is seeing that it's ok to be treated like dirt, that it's ok for boys to cheat on her and then blame their actions on her. It's not right and she's not going to know any better unless you show her.
Good luck in your decision. If need be, go and see somebody, maybe a counsellor, to talk this through. They may be able to show you another way of dealing with situation but you won't know unless you go and see.