I think you probably desperately want to think that your H is not of sound mind and that this has been the reason for his behaviour during and after the affair, but while I think some unfaithful people have emotional breakdowns after an affair is discovered, even then it is a breakdown associated more with how their lives have been wrecked by their own behaviour - and not deep sorrow and regret for the pain they have caused everyone else. It can be a very self-indulgent way of getting sympathy from those who have been hurt - and at worst, a way of deflecting responsibility for the carnage enacted by their own hands.
You will actually do more harm than good by being the sympathetic nurse-maid here, because if your H is genuinely suffering from a mental health crisis, he needs care from professionals and his own family and friends. Your children will be affected the more time they spend with him and you should shield them from that. They realise far more than you are perhaps reckoning with and the hugs they are giving you all the time are probably their confused way of seeking reassurance from you, because they sense things are not right within their family.
It doesn't mean you have to be unsympathetic, but you should stand back now and put your energies into creating a peaceful home for you and your DCs while your H stays away and fights his own demons.
If you look at your reactions since the affair was discovered, you have had a tendency to focus on anything but your H's culpability for what happened; the apparently volatile OW, the disapproving employers if the affair had been discovered and now, your H's mental health. I'm sure you've only done that because your H has evaded responsibility every step of the way and unfortunately you've accepted that holding pattern.
It is now high time that you stopped letting him evade the responsibility for his behaviour and got tough.