So, he is gaslighting you, has been far too close to another woman (which has been noticed!), invited her over for a play date when your kids weren’t there and was having walks in the woods.
On top of that he works 2-3 hours a day and does fuck all round the house- including child related care and bedtimes?
Plus now he isn’t talking to you (which is abusive) because you called him out on his shit behaviour.
If I have this right - What the fuck are you doing with this loser? What is he bringing to your life?
I am hoping you can see from these replies what a twat he is! I would not put up with this I’m afraid - if he hasn’t had a ‘physical’ affair it’s certainly been an emotional one. So look to work on you, because you deserve better than this - you deserve to be with someone who is respectful and listens/acts on your concerns.
If you are going to stay with this self centred, nasty man, you need new boundaries in place quickly
He needs a fucking job - as a priority - not many of us like working but thats tough.
While he is looking:
I would only cook for you and the kids going forward
I would expect him to be doing 80% of the housework during the week, and I would expect him not to talk to the woman away from the school gate.
These are non negotiable, if he wants you to save his marriage.