For the tax, as I understand the rules, the 50% is reclaimed from GOV.UK, it is to be treated as income and businesses are expected to pay VAT on that income back to HMRC.
It would therefore follow that the portion paid by the customer would be the only amount on which the customer pays VAT.
IE - VAT on £10.00 plus VAT on any amount they spend on top of the discounted deal which is capped at £20.