They don't really pay the money to the tax department on your behalf. They just pay a lump sum that covers all the wages that they have paid. They should be able to short pay the tax department next pay period (or however often they remit the tax) and adjust their internal accounts.
It all gets sorted out once a year when they tell the tax dept how much money you have been paid.
I reckon it's easier for them to try to get the money from you rather than sort it out themselves.
Nb. this is just my opinion, not sure of the ins and outs of the UK tax system, but they should sort it out, not have you do it.