Don't make ASSumptions. You don't know that. He probably DOES want to be there, at least so that he can be sure his new baby daughter is safe and his kids are cared for. And he probably cares about looking like a good guy. And yes, it's probably convenient that he doesn't have to figure out where to live today.
You need SOME HELP of SOME KIND right now. If you have someone else you can call that can come help, then do that and tell H you've got it covered. Or if you think you'll be ok at night so long as he takes the kids in the day so you can nap, tell him he can't stay there but you would welcome him coming for a few hours during the day so you can get rest.
You have to separate out your irritation with him, your desire to push for a confrontation (in the vain hidden hope that he'll blurt out "but no! I love you and want to be with you forever, I was wrong!"), and your absurd resistance to accepting help from anyone. These are all separate issues and need to be treated separately.
Also, remember to keep in mind, this is not a guy in MLC, this is a serial adulterer, who made you his OW without your knowledge, who is likely going to repeat his pattern again and again through his life, and unfortunately cannot be the husband you deserve without a lot of therapy and help, which he's not too likely to get. Just use him for the temporary help you need and let go of the rest.
