They are very unlikely to stop your Income Support. There is no evidence that you have deliberately defrauded them.
What might happen, as someone else said, is that they will decide that you received Income Support for a period of time when you were not entitled to it. They will say that that was an overpayment and you have to pay it back. Usually when you have an active claim they take some of your money every week going forwards and give you the remainder, until it is paid back. If you don't have an active claim you can arrange a payment arrangement with them, based on what you can afford.
It is good advice to write down the sequence of events, dates, etc as fully as you can. It will help to have notes during the phone call because if you're anything like me your brain will go blank when you're flustered! There may not be anything you can do about the period of time where, on paper, it looks like you were in a couple, so you might end up with an overpayment.
But just tell them the truth about it all exactly like you have here, including his abuse and your mental state, if you can, and it is very, very likely that your claim will continue.
If you are able to get a letter from your ex's family confirming when he moved in with them, that would be useful proof, or you could ask the Housing Benefit office at the Council to give you a copy of the information you gave them at the time. You may have to put your request to them in writing.
Going through all this must have been so hard for you, but in this instance, going into refuge will really benefit your case. That fact alone is extremely strong evidence confirming your situation.
I would advise you not to even think about looking for a job at the moment. You need some time and space and peace, in your new home, with your children, to begin to recover.
There are some brilliant videos on YouTube, made by a woman called Lisa Romano who calls herself the Breakthrough Life Coach. She is also a survivor of narcissistic abuse and her aim is to guide and support others to understand and heal. I found her really helpful in some really dark times and still watch her regularly xx