Welcome, as you can see from what the others have shared there is a ton of good advice in the healing library, start there.
You need to do a few things to protect you, and you kids though.
1. See a lawyer, find out what your rights are, and how things would play out should you go down the road to D.
2. Get STD tested. You don't know what he has been exposed to at this point, and need to protect your health. If you are letting him stay in your home, and having sex, then DEMAND he get tested as well, or he gloves up any time that you are intimate.
3. Make sure you are eating, sleeping, and staying hydrated. Seems obvious, but can be very difficult for a person who has just been thrown into this shitstorm. If you are having trouble with these things, talk to your Dr, and get a little pharmaceutical assistance.
It's really up to you at this point how this goes. You get to say if you want him or not. You also get to lay down the terms of what he needs to do if he is staying. The first of which is to establish and maintain NC (no Contact) with the OW (other woman). Nothing fancy, just it's over, I'm working things out with my wife, my family is my priority, please do not contact me again.
Next you need to let her partner/spouse know what's going on if she has one. A's (affairs) thrive in secrecy, and blowing the secret wide open helps to stop things.
Be prepared for him to break NC, many WS's (wayward spouses) do. It often takes something very drastic on the BS's (betrayed spouses) part to really wake them up. You cannot nice him back into the marriage. This does not work. Ask any of us.
I also want you to know that no matter his excuses, this has ZERO to do with you. He didn't do this because you weren't pretty enough, or smart enough, or fun enough. This is all on him, he is broken, and had crappy boundaries, and coping skills. He has to fix that. You can't fix it for him.
Deep breaths, and know that you will survive this, no matter the outcome, you will be smarter, stronger, and a better mom for it.
((((and strength))))