can't you put some into the children's Child Trust Fund?
You won't be able to touch it, but it will be tax free, and (hopefully) gaining interest until they turn 18.
Not sure how that would affect benefits. There's a limit on how much you can put in each year, but with 4 kids, you may well use up a lot of it that way. Also, buying some 'necessities' for your house could be good - could you take them with you if you moved later? Although you say your house isn't great, it sounds like you want to stay there for a while.
If there's any money left, you could keep just under the amount you're allowed to have ina savings account, the kind where you have to give 3 months notice, so that if you find somewhere nicer to live, and want to move, you have some cash to pay a deposit/buy things for new place?
20k sound slike enough to stop your benefits, but not enough to buy a house, start a business etc.
There MAY be some way of legally transfering the money to your dc's without you touching it. If it is put in trust for them, so you can't use it, it may well be tax free & not affect benefits. An IFA should be able to help you with this.