First of all, it is extremely unlikely that this isn't a full affair. People don't usually walk out on their marriages if there has only been kissing and flirting.
Please stop thinking that this is your fault or that you've caused it. Your partner had a responsibility to protect his fidelity and to talk to you once it had been threatened. He did not. He only admitted anything after it had happened. Right now he is desperate to share the blame because he knows that having a secret relationship is indefensible.
The stance you're taking now is unlikely to work if you want him back and for him to remain faithful if he does.
Instead of thinking that you're partly to blame and that you've caused this, it would help more if you got angry that instead of discussing any grievances, your partner has started a relationship with someone else and has walked out on his responsibilities. There is no excuse for that.
So your best bet is to tell him that you need some thinking time because you're not sure you want to be with someone who is unfaithful to you - and you'll take a lot of persuading that you can ever trust someone who cheats and then tries to blame you for it.
Don't let this man think all the choices are his. It's very likely that he's having an affair and won't want to come back to your marriage until it's over, whether that's while he's under your roof or apart from you.
The only thing that ever works in this situation is loss whereas giving people time and space to make choices never brings them back, or if it does they will do it again to you because there is reduced respect.