dear Anthony, welcome to this site.
From what you are saying is that you appear to be caught in limbo here by your wife, and sorry to ask but is the flat jointly owned.
She has done the dirty on you and the kids by living with someone else, so I wonder who has jurisdiction on deciding who can live in the flat.
The other option is to settle this mess and get a divorce which would mean to sell the flat in which you can then go your own separate ways, this is going to be heartbreaking for the kids, but it seems as though the marriage is over, this is never pleasant for the kids when it happens, as I have experienced myself.
This is going to be most difficult as it seems as though she will dominant this but your own sanity is important, so you have a few problems here, the safety of wanting your kids, not to be controlled by your wife, and mostly your own mental health, which is very important.
You say that you seeing a counsellor which is good so I wonder if it is sort of helping at all, and whether you should be on some medication as well.
We always ask people if they are taking AD's because sometimes it works and sometimes people struggle with taking it.
Can I ask how old your kids are, and hope to hear back from you. Geoff.