Hi, I'd like to help you with your question.
From what you said, it sounds like your husband had a relationship with this woman, possibly had a child with her, then continued the relationship off and on, even while married to you.
There is no way to tell if he is being truthful at this point. You can ask him to get a paternity test, but if he is not willing and he is not paying child support, there is no real motivation except his willingness to do it for you, which he should for your marriage and the trust issues this has created.
His expectation that you treat this relationship as if he never cheated on you is unrealistic. There is no way for you to pretend the situation did not exist. He may chose to do this, but to ask you to do it as well says he is not dealing with the situation as he should be.
I highly recommend you both seek counseling together. You need the support, and you also need to talk this out with a neutral person so you can be sure that the topic is equal and fair. You also need to work on trust. If your husband is just informing you of this now and had a whole history with this person and possibly a child, you needed to know about it. Since you did not, this becomes a serious trust issue and trust is vital for a relationship to continue.
He also needs to reassure you that he will discontinue contact with this woman. Since the child is an adult, he no longer needs to have contact with the mother. He can develop a relationship with his daughter, but you must be included and he must acknowledge you as a priority in his life. Although that does not mean you get included in every contact he has with his daughter, you need to be informed of it and kept abreast of the relationship. This is also to help rebuild your trust with him.
I hope this has helped you,