I think it’s down to what happens next and sadly most of that lies with the ability of the cheater to analyse themselves and be honest with themselves and their loved ones - not something they tend to be good at. They tend to be compartmentalisers and liars and attention seekers who require validation. Many have a black hole that needs constant validation.
Do not entertain any counselling if there is a hint at ‘your marriage caused this’. No it didn’t - if it did anyone who was unhappy would cheat. Only some people cheat. You do t want to leave counselling thinking it was your fault.
This is an issue of poor boundaries, lack of self respect and integrity. So why did he cheat? What reason did he give? Any hint of anything to do with you and cut him off. You didn’t cause this and you can’t stop it. This is down to him. You cannot let your self esteem be destroyed so cut him off at any such nonsense. He stopped being able to blame the marriage when he hid the first message. If he was unhappy he should have said before he hid the text. Cheating will never resolve marriage issues it only makes them worse. If you were the nastiest wife in history he could have left and kept his integrity and stayed true to his vows - he chose not to - that’s his issue not yours or your marriages. You can no longer fix the hole in the roof as he has put a bomb in the building and blown it to shreds. You need to start from scratch.
Until he has done hard work discovering why he allowed this to happen you can’t work through it as he isn’t safe. ‘I’ll never do it again’ doesn’t work as it doesn’t make sense - why won’t you do it now when you did it then! Your marriage is now going to be battered by the emotional fall out from his poor choice - if he thought it was bad pre affair he’s in for a nasty surprise as it’s about to get 10 x worse. So ‘I won’t cheat’ doesn’t cut the mustard.
What’s changed - you catching him doesn’t cause change (just hiding stuff better).
Does her partner know? If not tell him now - makes it harder to go underground if both of them are fighting to save their marriages. Also he may find out different details.
I also hate to say it but when you dig around you are highly likely to find more. Be it dating websites or previous emotional affairs or web cams or chat rooms. Do not leave a stone unturned on all devices. Get him to write a timeline and do a lie detector test (you can get UK ones) if you even remotely believe in them working. People on the SI website say they lead to car park confessions.
It is your decision to stay and I understand why you want to. But you cannot overcome or forgive what you don’t know. And he needs to know why he did it or he is not safe for you. Also expect there to be more to come out via trickle truth.
I hope you are okay op. Get counselling for yourself - the only gift you will get is rebuilding who you are and maybe finding parts of you that are lost. You are between a rock and a hard place, collateral damage to his poor choices and lack of integrity.