Cheating (long message)

So I've been with the same man for over 4 years. We have twin toddlers together and a third that just got here. We have always struggled with mental health and not getting the help we need. ( For anxiety and depression ect) during my first pregnancy he was interested in someone other than me and I almost left him. He promised to change but he never went to get on medication that would help his mind operate correctly. His family has a long line of bipolar and mood disorders and I know disorders of such can affect relationships. We have had rocky times and I really struggle with depression during pregnancy. We started to have financial trouble and things started to get worse and worse. We were just hanging on because of our kids and the relationship felt over so we sat down one night and had the conversation of " we both feel this is over unless we change something" from that night I started to act on the changes. He asked for privacy on his phone during this time which I gave him because I just wanted this to work, I was 9 months pregnant with twins toddlers already. He began working 70 plus hours a week. We stopped seeing each other a lot. Then I gave birth to our baby. He stayed in the hospital for the birth then went home. I felt in a time where I really needed him I was alone. For three weeks after I have birth it was the same thing- he would work from 12pm-1 or 2 am and then he would sleep until 10 and get on his phone in the bathroom as soon as he would wake up. He would go into work on his days off. On one day off he finally stayed home the entire day and he fell asleep and took a nap. At this time I was still working on our relationship and thought things were getting better, other than he was working so much. I begged to see him more but if he had to work that much I was going to support what he needed to do. During his nap he got a text and I didn't think anything of it. I hadn't checked his phone in over a month. I glanced at it and saw hearts and a sweet message. She said she was sorry that she slipped up and accidentally told him she loved him. I confronted him about it and he tells me it was nothing to him. That they kissed and made out and texted about other things but no other things ever happened. That he stayed after work only a few nights and talked or made out with her. These nights while I was home taking care of his three kids working on what I thought was a relationship we both wanted to save. Now he wants to work things out with me and I'm just not sure if I have it in me after all of the events happened. I need advice. I don't want to lose my family but I also don't want to lose myself.