Shitty situation of course, but... First of all, and remember that for future, your husband/partner/boyfriend does not ever seeing ex-girlfriends! You can trust him as much as you want, but things happen and feelings coming back and kissing (and more) coming. Even if he loves you, it can happen.
Right now, it depends on what do you really want. Think about that. Think about yourself, not children, not marriage, just yourself. Do you want to save these relations or separate? Can you forgive this? But for real...forgive like forgetting about it forever and never have thoughts about it. It's important that feelings which you have now won't be eating you from inside later because now you'll block them for sake of marriage/family/kids/whatever. It will take time to decide, so I would advise next:
You have to make him understand that its not something minor like he stepped on your foot so "sorry" is enough, words are not enough, he has to DO things, not just say. So talk to him first, calmly. No Italian scandal because he will be happy to run away and no tears because he just will feel sorry for you. Thing about your dignity, your self-respect and your love to yourself, and with all these feelings talk to him. Tell how you feel about that, tell that he did hurt you, that you are not cool with such things, and that you want a time to think about and make some decisions. Here conversation finishes. No discussions, no arguments from his side. End.
Put the deadline for thinking(month should be enough) and kick him out, don't worry about where he will go: friend/hotel/hell - not your problem, he will handle it. He has to know that its a big mess up and with you its not allowed like that. As you have kids you will be seeing each other anyway, thats ok - see him, don't abandon him from children, but be cold (!!!), friendly, happy, polite and cold (don't forget to be beautiful;)), you are thinking about important stuff so no hugs, no kisses, no chats. Probably you would like to finish that earlier, but DO NOT! Keep your word and wait a month.
During this month you need to kick him out of your head and this situation, thats difficult, but do as much as possible. Think about your relations, are you happy, do you feel in love with him? Do you want to spend the rest of your life like that? That time should be enough to realise your feelings without his impact and pressure from that situation. Also do things for yourself, like start some hobby, go to spa, buy beautiful dress, do whatever makes you happy. Do you feel better without him than with him? If you decide to separate, it's ok, don't worry about kids, they will be fine and there are plenty of men outside, so you'll get another one. Meanwhile your husband, if he wants you back, should start to act and doing things for you, he has to deserve your forgiveness. If he won't DO anything then its probably time to let him go. Man has to fight for his woman and if he doesn't then he is not your man.