It sounds as if they are a dealer - in the business of buying and selling cars. It is essential to you claim that you are able to prove this because, if they are, the Sale of Goods Act 1979 implies terms in to the contract. If they are not then there are no such terms implied.
I answer on the basis that they are dealers.
If the car was not of satisfactory quality then you could have gone back to them requesting that they remedy the repair under s14(2) SGA 1979, if they could not repair it then you would be be able to ask for a full refund.
The difficulty is that unless you have given the dealer the opportunity to repair the car and remedy their breach and they point blank refused to engage with you then you will not be able to claim the cost of the repair that you have paid for by the other mechanic.
Following repair by the mechanic if the car is still not of satisfactory quality to the extent that you are denied substantially all of the benefit of the contract then you can go back to the dealer to request further repairs of a full refund of the purchase price you paid for it. If they do not comply then you will be forced to issue a claim at Court to recover the monies, but before doing so you may choose to take specific advice from a local litigation solicitor so that he can write a letter before action to them to see if this prompts their compliance.
You can find solicitors in you area by using the Law Society's search engine, enter your postcode and "civil litigation":-
It's a great shame you went to the mechanic first I'm afraid, sorry it could not be better news.
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