Also op, please learn to take personal responsibility and accountability for your actions and grow from these things.
The both of you got into this mess together.
You knew that he was married, without any proof or investigating you automatically believed him, and gave yourself to him.
Chances are you didn't protect yourself.
You should have left him alone.
You knew that he was married... regardless of what he said.
If you have these children, you are
Trying to stay connected with him forever by intentionally tying yourself to him.
You know that anytime he changes his mind he can file for joint custody and be granted it.
Anytime in life, he can decide he wants to be a part of their lives, and if the kids want him there if you're a good parent, you'll let him...
At anytime, when older, the kids can look him up or he can look up the kids contact them perhaps lie to them about you keeping them away from him, and they'll believe him and perhaps dislike you..
They'll believe him because they would have heard only one side of things their whole lives.
If you have these children, the other child is their sibling. What are you going to do about that??
I think you are still hoping for a chance with him because you may love him.
I think by having these children, you'll feel like you have a "forever" unlimited chances with him.
I don't think the wife knows and he has fabricated everything and is willing to give you "hush" money.
If she knows, he's a serial cheater and he's done this many times before....she just stays and tolerates it.
He's a serial cheater and knew his wife would stay...that's why he was carrying on with you so carelessly and reckless..
He wouldn't lose anything if caught.
Your self esteem isn't high or non existent.
Please learn to love yourself and raise your self esteem and self respect.
Perhaps therapy would be beneficial.
Whatever your choice is, learn and grow from it.