The problem is always the innocent children. Decent humanity is what stymies things in this sort of situation. But then nothing changes, nothing gets better.
Perhaps the answer is to set a low limit that should just cover the needs of a small family who are temporarily on hard times. If the children are then seen to suffer because either the money is not being spent wisely, or because the parents already ensured there were too many to be covered by welfare, then the parents could be charged with neglect of those children, be given a gaol term.
The children would need to be taken into care, after all they are already being funded. It'd cost more and not be ideal for the kids, but the responsibility for that experience would be down to the parents, and public finances may improve if it acts as a deterrent in the long term to expecting the taxpayer to fund the irresponsible ?