Thanks for updating VB. This must be a horrible, shocking time for you.
I have a feeling you're pinning too much on the counselling, or at least couples counselling. What might be much more apposite is solo counselling, for your H. There is nothing in your posts to suggest that this problem is triggered by relational discord and everything to suggest that this is his personal problem.
What has he said he is going to do about this relationship with the OW now? Is he going to end it with her, or is he ambivalent? You need to know much more about this relationship and the OW herself and while I understand that your focus last night was him and your relationship, it's interesting that she wasn't mentioned by him much either.
Sometimes that happens because the H wants to keep the OW and the relationship in a kind of bubble-wrapped shrine; unsullied by the comments of mere mortals. Sometimes it's because the unfaithful party doesn't attribute a huge amount of relevance to the OW herself and knows that it's more about the adventure she represents, than the person she is.
An objective onlooker can see that this OW is not particularly special, because almost anyone would have fitted the bill for a man who likes new beginnings and the thrill of a new adventure. However, you need to know how your H is seeing this. It is good that he at least has the presence of mind to acknowledge that bright beginnings tend to fade in time and become normal relationships.
From your point of view, having had this chat, don't neglect yourself and your own needs. Don't be afraid to attach some conditions to him staying in a relationship with you. He really doesn't hold all the cards here. A minimum condition for your safety is that he ends this relationship now and in the longer term, gets some therapy to find out why this keeps happening.
If you're still insistent on couples counselling, I'd do some research of your own, double-checking the apparent lack of appointments at Relate (but some centres also do telephone and online counselling) and speaking to private practitioners yourself. However, as with all counsellors, choose wisely and ask questions about their experience of infidelity and especially a person's individual vulnerability to it.