You were separated but agreed to stay living together for now until one of you wanted to start seeing other people. He wasn’t seeing someone else, he had a ONS, which he was fully entitled to do, given you had separated. He should have told you before you had sex with him again, due to the risk, but not due to the relationship. Stop feeling like he cheated on you, because he didn’t. He was sexually irresponsible having sex with you without informing you he’d had sex elsewhere, but that’s all he ‘owed’ you.
You need to park this and think about the actual relationship, why you separated and either try to make a go of it with him, or separate properly. Stop kidding yoursekvesvtgat this dumb ass living arrangement is ‘for the kids’, it’s not. It will damage the kids, not help them.
If you REALLY think he is able to be the man you want, then give it a go. If you actually know he’s not capable of it, pull your big girl pants up, and tell him it’s over.
Sorry if that sounds harsh, but sometimes harsh is exactly what we need.