Sorry you're in this mess! Your situation smells bad to me.
I feel like my WH still has far to go but one thing we successfully changed after D-day was that we start sleeping together, in the same bed, during sleeping hours, at the same time. If he wanted to watch movies, etc., he could do that anytime, but we were married and that's what married people do. This is a low bar. I insisted on this from the start, though I was still swirling from the discovery, but I felt like I staked my claim as his wife and protected myself at the same time because I knew what he was up to during the night.
Go with him, with crying kids in tow, next time boss lady needs a ride. Otherwise, she can get an Uber like the rest of us.
This woman is his boss so it's not necessarily bad that he brings her up in conversation. But is this normal work stuff or does he talk about her personally. Is there ever any criticism of her or the usual boss bashing? Or is it all admiration and chummy talk?
I personally make accidental calls on and off, but my kids say this is a because I'm not so savvy using my phone and that young people rarely, if ever, do this.
Somehow you need to get some sleep so you can think clearly. Alone or together, you need to be strong, mentally and physically. Don't be the victim in this (even though you are, of course) but you'll be better off if your can take care of yourself and fight for your place as his spouse and mother of his children, or fight for freedom from your betrayer. Don't stay in the middle. I say this from experience. I, too, had 3 young children and looked at leaving. I didn't know of infidelity at the time but mine was not a supportive husband. I wish I'd been stronger, not a doormat, that I'd defined what I expected and needed from him at the time, got it, or just left him (which would have been the best choice.)
Love your kids, get help, get healthy, know that you can do this. Unfaithful, selfish men do not make good husbands or fathers, so don't stay for the kids. If he's screwing around again, leave him. Regardless of any infidelity, if WH doesn't step up after cluing him in about your needs, leave him.
Know that a better life awaits you after this. Suerte.