Another person who has read this, just wanting to add some words of support.
Don't make any hasty decisions - reserve the right to take as long as you want to thinking things over, taking time to let the shock subside and for things to sink in. Reserve the right to ask him to move out to give you space - it's not an unreasonable request. Reserve the right to change your mind on anything at any time. Make sure he knows it - during your discussions from now on in, he should be the one walking on eggshells not knowing what's going to happen - not you.
Any hint of blame, excuse, any pushing of any responsibility for this onto you or 'the state of the marriage' (i.e. code for your responsibility/problem too) is utterly false, and you shouldn't tolerate it for a minute. If he even breathes a HINT of this attitude in any dissection of this, stop discussion instantly and point out to him that no matter WHAT problem he perceived to be there, he ALWAYS ALWAYS had a choice of what to do about it:
- Talk to you and try and solve it - thus prioritising the protection of his relationship and family, and keeping his marriage vows;
- Have an affair - thus prioritising his own selfish, indulgent agenda, putting his wife and family last, and breaking his marriage vows.
His choice! Make that crystal clear. HE MADE THE MARRIAGE BREAKING CHOICE.
As a follow up to this, make it clear that any discussion about what to do is ultimately a discussion on whether YOU can be convinced that this lying, cheating, FAILING husband is worth continuing with, or whether the family (you and DC) would be better removing him from the immediate family unit. It's NOT a discussion on how 'we' can make things better - at least not until much much further down the line. If he wants to convince you that he is worth not sacking, he needs to start with complete, honest disclosure on the affair (no, he does NOT get to decide what bits are 'unnecessary' or 'not important' for you to know), and complete willingness to put right the issues WITH HIS PERSONALITY which led to this failing in his job as a loyal husband and father.
Of course, you may already know in your heart that it's over, that you simply don't want to continue. That is absolutely FINE. You have the absolute right to say 'I'm sorry, but a rebuilt marriage with someone I know to be capable of cheating on us (yes, it's all of you he's cheated on) is simply not what I want from my life'.