if he cheats once he is going to do it again. next time it will only hurt you and your children even more. they may want him around now but you will be able to find someone who will never cheat on you. Why stay with a man who makes you unhappy and cheat yourself out of real love. go find someone who makes you and your children happy!:)
In my honest opinion I don't think you should stay with any who cheats and you are feeling unhappy but it seems like you two have been together so long and you have kids and they are a priority that maybe trying couples therapy would help your relationship. There is a chance issues could be resolved if you two took the time to let everything on the table and say "this is it". You could find that it is better to stay apart and share your children to stay away from the hurt to help you heal
I hope that I've at least given you an idea of what to do.
As the kids grow older, they will understand. Because you have to trust the man, or woman your with. They will still have contact with him. If you keep him around and stay married, will you really be any happier? Second guessing every time he goes somewhere. Wanting to check his phone, and heaven forbid the whole likelihood of him introducing a vd. Respect yourself, It doesn't help your kids to be around this, what if they start to think it's no big deal like you think? Trust you are doing the best for them, and for yourself. Life will be tough but only for a little while. I wish you all the luck and your kids too.
Perhaps you and the kids could live nearby -- close enough for them to walk back and forth so that both of you look after the kids but live apart. Social workers can help you plan the most practical and least upsetting transition.
your kids may want him close but, your teaching your kids that that its ok to stay when they grow up they will be compelled to stay also and, look at you poor thing, you have 1very short precious life, you know you deserve happiness right? go, leave, be happy teach them that no matter what, they will have both of you and show them how strong and independent you can be.good luck and God bless.