Batgirl, please, this is really important.
You cannot let him stay even one night.
If there's any chance you don't want him back - and there's obviously a massive chance of that! - you may need an Occupation Order to keep him out of the house.
Right now, that's probably one trip to a solicitor. I'm not a legal expert, but he's clearly moved out, he can't claim it's his home / residence.
You let him back in, for ONE NIGHT - and who knows?! You might fuck up your chance of an occupation order.
How many threads have you seen on here where people are desperate for an ex to leave, and they won't???
Your kids must NOT wake up to daddy re-appearing. Don't put them through that.
Text him NO. Tell him that you don't want him moving back in.
You know he probably had sex with the OW last night, those clothes he is bringing back were probably washed by her, he'll come in smelling of her as 2 hours ago he was in bed with her?
And he's only coming back until he's "sorted".
Oh love - SELF RESPECT!!!! He doesn't get to treat you like this.
What an absolute fucking arsehole he is. Not even pretending he wants you back, either.