Well, let's perhaps look at it this way. You own the house, you pay the mortgage, the kids live there, so by paying the mortgage and allowing the kids to live there, you are, in effect, providing for them.
The central issue, I think, is what happens to the house.
May I assume you are divorcing ? If so, the Court will be concerned to ensure there is a home for the children. But it is important that there is a home for you too ; and it is clear that running two homes is putting your budget under great pressure.
But the problem is that you have a statutory obligation to pay child support. Now if, for argument's sake, your wife had the right to live at the
FMH for some time to come - perhaps till the children have grown up - then she will normally be expected to pay the running costs out of what resources she has or might acquire. These might include maintenance ( child and spousal ) her wages, if any, benefits like tax credits, child benefit, council tax benefit and so on.
I can't really get any ' feel ' as to whether that could be reasonable, but I suspect that, as long as you keep on paying the mortgage, and she lives there, she may well be doing rather well - at your expense. She may have very little incentive to disturb the status quo.
In theory you have an obligation to pay child support : I suspect no obligation to pay her maintenance ; but as sole owner you have an obligation to pay the mortgage. It may be an option for you to discuss this with the lender and see if they can help.
May I ask whether you think your wife might be willing to discuss this issue ? You will have to pay child support, you may have to pay spousal on top of that, and if your wife can't manage then it is unrealistic to expect you to pay rent and a mortgage while she pays nothing.
LMM