Hi Bugsy, so sorry to read of what you are going through right now.
My DP had a one night stand early on in our relationship. A bit different from what's happening to you - and we had no kids - but the deceit and hurt was still there. He allowed her to keep pursuing a relationship by phoning him but eventually admitted what he'd done to me, some weeks later.
Of course he SAYS it was a one night stand but I have no way of knowing for sure, and I urge you to be persistent in seeking the truth from your other half. Typical reactions in the Rumbled Male are "It only happened once", "It meant nothing", "She's moving/she's left the office/she's already ended it", "We were both drunk, it just happened" etc, and I'll bet in most cases they are lies generated by panic. I still wonder if my aghast reaction to the news put the frighteners on my DP and stopped him telling the full truth.
I made my DP call the other woman - in my presence - and explain that he was actually already in a serious relationship and would not be seeing her again for any of the dates she had been suggesting. He was VERY reluctant to do this (wonder why...?) but I forced the issue and it felt better knowing for sure that he had put her straight and ended things. I heard exactly what he said, so no "...I'll call you in a few weeks when this has cooled down" type conversations. I was also able to gauge her reaction; she accepted it for about two minutes, then called back and let rip with a stream of abuse. As far as I'm aware that was the end of their contact. I think it would serve you well to try a route something like this, to be sure that your husband is sincere about ending his affair and focusing on his marriage - and not just temporarily.
Most of the rest of the stuff can be worked through, IMO. You clearly want to repair the damage, ideally, and I decided that too, as I loved the guy and had a great time with him, and saw a great future with him. I grilled him for info about every detail of their night together and he was very honest about things, which I appreciate. I knew the girl, he told me that she had mentioned me to say - in a variety of ways - that I was ugly. He also admitted that he had denied even fancying me, let alone having a relationship with me. Both those things really hurt. I also demanded to know all the sexual side of things and he told me, which must have satisfied some twisted curiosity I had but all the same was very upsetting, and for a long time afterwards all I could think about during sex was them.
All this for a one-night stand with a relatively new boyfriend...I don't pretend to understand the depths of your hurt. Myself and DP have had a great relationship and I'm glad I surprised myself by giving him a second chance. As far as I know he's never done it again, and no-one in any relationship can be 100% sure of fidelity unless they are with their partner 24/7. However it does get worse before it gets better. I used to follow DP in my car when he went out at nights to check he wasn't going to this girl's home. I used to write down all unlisted numbers on his mobile and get my brother to call them anonymously and then hang up, to see if they belonged to women. I still check his phone for messages and new numbers, and although we joke about it, there's always a more serious undercurrent. I certainly wouldn't stick around if anything like that ever happened again, but to be honst, even if it did, I wouldn't regret sticking it out the first time, he's a really decent human being, and everyone makes mistakes. Yes it's a pretty big mistake to make, to jeopardise your marriage and your children's happiness, and to cause your own wife such pain. But the situation is not insurmountable. What I think is crucial is the guy's willingness to admit his wrongdoing; so many make such spurious excuses, like they work so hard, temptation is always there, the wife doesn't understand me, she's always busy with the kids, we don't see each other enough, she's gone off sex, I get lonely away from home, she was coming on to me so strongly I couldn't resist, my mates egged me on, I never have any excitement in my life, blah blah blah. It's pathetic. My dp tried the old "I was drunk" story but he also had the decency to admit that he was a guy who tended to follow his penis around, insensible to the consequences, and that he was oblivious to everything except getting into this girl's knickers. I'd say in your case your husband has to tell you the full truth, and then grow up a bit - travelling a lot and working with flirty colleagues is just life, it's what we all do, not an excuse for ruining your family. If he doesn't accept that he made a big mistake and a very stupid mistake then chances are he might fall straight back into the same situation in either months or years, and justify it with the same idiotic reasons. Make it clear there will be no more chances.
Sorry to have rambled on but I know how harsh this feels when it happens the first time; even though there's a big difference between your situation and mine the pain's still massive. I really hope this is resolved in the best possible way for you, do take good care of yourself.