In my experience HMRC are quite understanding of this situation, but I can see why they are questioning why you are still paying council tax as a couple if he doesn't live there.
One of the first things I did when he left was to inform the council that I was the only adult in the house so that I could claim the single person discount. I also too my ex off the electoral register.
He moved in with his girlfriend, but didn't change his address for things like his bank statements and driving license, despite my requests that he did this.
6 months later HMRC contacted me with their suspicions that he still lived in the
FMH and that I wasn't entitled to WTC.
Apparently, HMRC have an arrangement with a credit reference agency whereby they send all details of single claimants and ask if there is any evidence of someone else living there.
I contacted HMRC and explained that my ex had moved out to his girlfriends and that I had asked him to change his banking details etc but he had not done so.
They were understanding and said that they believed me and the case was closed.
When you contacted them, didn't they ask where he was now living if not in your home?
The fact that he is living somewhere that he shouldn't be really isn't your problem and I think you will have to come clean on that
Regardless of the fact that you had an arrangement with him that he would continue to pay all the bills, the facts of the matter are that you were entitled to those tax credits and you just have to do whatever it takes to prove it. The CAB is perhaps your best place to seek help with this.
If you have contacted them,