I agree, stop talking to him about this, he will never agree. You are eating your energy.
This is what I would do.
Stop talking to him about it. If he says anything, say that you’ve thought about what he said about it just being a friendship and you are going to work on your own issues and get some counselling.
Book yourself some counselling sessions now. You need to tell the counsellor what you’ve said here and they will help you work out what YOU want to do. Maybe another RL perspective will help you see that your own anger and hurt are normal and legitimate and that your husband is lying through his teeth.
Your husband has made the choices clear . Stay and put up with his affair or leave.
Which one do you want ?
Every single person on this thread is advising you to do the second - leave ( as in divorce him - you are the main carer and have young children so he needs to leave ).
But you have to make this decision in your own time. And I also suggest that you plan this very carefully and get legal advice and all paperwork in order BEFORE you tell him. If that’s what you decide to do.
Am I correct I’m thinking that you are 43 and have primary aged children and you are legally married ? I mean not just a religious marriage or what some people call “ common law“.
You say you are getting “ state money “ - Do you mean benefits as neither or you work ? Why do you think this would change much if you divorced him ?
Is there any reason that you couldn’t get a part time job once you left him ?
Things are maybe not as bad as you paint them. He would have to pay child support for his children ( unless he had then 50% of the time ).
Remember you are doing 100% of everything now that he has left you.
BTW do NOT allow him to come and see the children are your house. Insist that they go to his mothers or he takes them out somewhere.