Thank you for contacting JustAnswer.
I am sorry to hear about the problems you are experiencing in your relationship. The fact that he has had an affair suggests that there are problems in your relationship that have not been addressed and/or problems in his life that he was trying to avoid. Generally, affairs are not helpful as they typically only serve to complicate matters further. They are generally only a symptom of the problem. They may also be a way to try to avoid feeling uncomfortable feelings and facing life problems.
It may be very helpful for your partner to begin individual therapy in order for him to learn to cope in healthier ways and to face his feelings without avoiding them through these types of behaviors. In addition, couples therapy can often be a way to discover whether the relationship is able to be fixed or not. Either way, it will take time for him to earn your trust back as trust is something that can be lost quickly but takes a period of time and trustworthy actions to earn. You may find the book, After the Affair
to be helpful in coping with this.
In order for him to be able to avoid cheating in the future, he will need to work on how to effectively deal with problems in the future without resorting to cheating behaviors. You cannot make him change. However, his being open with you about allowing you to check up on him is a positive sign. I would also wonder if he has a pattern of cheating. Past behaviors tend to predict future behaviors, unless the individual takes responsibility for their actions and works on changing. If he is to earn your trust back, he will need to show you through his actions that he is remorseful, (not just blaming it on excuses), has learned something from this experience, and is trying to change. It could be helpful for you to try not to make any major decisions immediately as you are probably not thinking very clearly right now. Give yourself time before you make decisions about this. I am glad to hear that you have started therapy to have some support in working through this. Please let me know if I can help further.