First please try not to think the worse.
There are still single claimants with tax credits whose exes still live in the same house so just because he is there at weekends does not mean that they are automatically going to treat you as having made a fraudulent claim, which does not appear to be the case anyway.
First you need to give the lady the facts tomorrow -
1) You are separated and have been since xxx date
2) that all the bills apart from the council tax are in your name, as this was tied in with the rental agreement from 2 years ago. But that you can provide evidence that you pay all the bills, and if she would like, you can send copies of bank statements and copies of bills to back this up.
3) State how things are with sleeping arrangements and eating and your general lives, and that he comes to see the children at weekends, and as he has no family, he has no where he can take the children, but if the tax credit office wish to treat him as still living at the property, then that would be a fair assumption, although you must give as much information, to support the fact that you live separate lives.
4) Offer any other evidence of how you live your life as a single person, especially at weekends, when he is there.
5) Advise her to wish to comply in any way possible, and ask her what other evidence would help support the facts as they are.
6) List as many social events that you do just you and the children, or just him and the children, especially family events, holidays and days out.
It would be better if he was paying CSA - so see if you can track what payments he has given you - and the fact these have been few and far between further supports the fact that this is not a relationship, but just two parents, one of which cannot afford to move out, but pays nothing towards the upkeep of the house or your life. Just support for the children to spend as you see fit.
Please do not worry - (and I know its easier said than done) as the truth will help you and this dreadful time
And the CAB are right in their advise