FFS get out sweetheart.
You're being gaslit from all angles on this one.
First by your shit counsellor (who has a vested interest in "helping" because they are taking your money)
Secondly by your shit husband, who yeah... got dumped when he went into shock over being kicked out.
I know you're here trying to find a way of accepting the situation, trying to find a way of reconciling, but listen to what your heart REALLY tells you.
This is not what you wanted, it's not what you want to deal with and I can tell you now, you are throwing money at a situation that isn't going to heal, it'll cost you your inner peace and you'll never trust him again.
I'm sure you have responsibility in the situation, I'm not going to say otherwise, but at the end of the day every single adult human being is responsible for the choices they make, and when you are married, you go to the altar, you say your vows, and if you break them then that is a choice.
His choice was to stick his dick in another woman, and then plan a future with her to boot, to lie to you multiple times, to ONLY want to come back when he'd been found out and discovered the grass wasn't greener.
This is one situation where I will squarely say filefordivorce. Divorce him.
Would you be heartbroken and have to build your life back up? Yes. Would it hurt like hell? Yes. Would you be depressed as hell? Yes.
What do you want though? A torrent of pain as a result of making that choice, or a very slow drip feed of pain for fucking YEARS of your life, wondering what he was doing on his phone on holidays, when he's working late, when you're just sat there watching TV on a Monday night?
Not only that, and I hate to guilt you with this, but you are SHOWING your daughter that this is acceptable, and your daughter will likely go on to believe that "well, mummy put up with it, so perhaps I should put up with it".
Counselling isn't going to fix it, not unless the counselling is from YOUR counsellor, not a couples counsellor but someone who has a vested interest in YOU and only you, someone who asks you why the fuck you want to stay with a person who has repeatedly lied, clearly only follows their own self interest and has betrayed you, your children and their own character.
Ending a marriage is gut wrenching, it's horrific, heartbreaking and soul destroying, but you're not the one who chose this path, he did. He has destroyed your peace, the sanctity of your home, and he's exposed your children to this as well. He's a piece of shit. He isn't worthy of the responsibility of a husband.
It's the repeated lying for me, and the fact he comes crawling back when his bit on the side ditches his pathetic ass. But that's ok, because YOU'LL take him back.
He's sorry he was caught, he's disgusted that he put HIMSELF in this situation, I don't give a fuck what he tells you.
Yeah i've been down the counselling road before after this kind of thing, and let me tell you now that no other human being can magically take this away because they have a piece of paper that said they've done a course. It doesn't make them God, it doesn't mean they have the ability to fix this for you. People fall into the trap of counselling hoping for a magic bullet, there isn't one. Now you're on here because why? The counselling isn't really working because you actually know the answer, you know damn well this is unacceptable, you know you don't trust him and you want to ease that feeling - well you can't. Nothing can for long.
Get out.