Well, you signed a contract and agreement with the company - and they signed it with you. So it's agreed and signed for.
So it may be the case you can simply stand by the agreement, even though they made a mistake - and they pay for making their own mistake.
However, there maybe something in the small print that says that can get arsey if they want to be.
So keeping the existing agreement may be more hassle than it's worth.
Probably best to ask at the Citizen's Advice Bureau for a proper second opinion on this.