Hi, hope you’re ok, just wondering whether you got some legal advice? Your house is in your name so you are the sole legal owner and it would be up to your ex to prove he has a claim on it. It is possible that he could, if he could prove that you purchased the house together with the intention of it being a family home, I.e. there was an implicit trust that you would both ‘own’ it in practice. However, if he left this home willingly then you can argue that he has therefore removed his interest in the property. The court will look at the conduct and intentions of both parties when purchasing the house. I am assuming you were not married?
You also remain there with the children, and he has a legal responsibility to house his children, so this will be taken into account.
You say he paid the mortgage - was this paid from his bank account? Can he prove that he contributed to the mortgage? Or if he paid you money, could that be for other things- presumably you are looking after his children?
If he did decide he wanted to lay a claim on the house, it would be very difficult for him to actually get anything out of it. He may be able to claim for a proportion of the proceeds of sale when the house is sold, but he would not be able to force the sale of the house to enable you to release this money until both children are 18 years old; plus, since the house in your name, it would be even more difficult for him. It may be that at the most he could get his contributions back minus costs associated with the children with statutory interest added. Technically if he was living in your house he should be paying some money towards the property anyway.
It is complicated and he could make some claim but I doubt he would get anywhere with it and if he did succeed, it probably wouldn’t be worth his while anyway. I am basing this on you not being married though - if you were married then it is a bit different. Xx