Hmm. A garage that offers a repair free of charge? Why would they do that?
Second hand cars bought from a garage still come under Sale of Goods Act. You have allowed the garage to remedy the situation by offering a repair. If the repair cannot be completed within a reasonable time you still have the right to ask for a refund.
This info from Trading Standards might be useful:
If the fault was present when you bought the car, you do have other remedies, even if you have left it too late to get a refund. You are entitled to seek a repair or replacement. These have to be carried out within a reasonable time and without significant inconvenience to you. If replacement or repair is not possible, then the law allows for the options of full or partial refund. Each case would be judged on its own merits.
The onus is normally on you rather than the trader to prove a claim, i.e. that the car is defective in some way. However, the law now states that if you are claiming replacement, repair, full or partial refund within the first six months of ownership, the onus is on the trader to prove that the goods were acceptable when they were sold. This is called the �reversed burden of proof�.
Contact your local Trading Standards office to find out if the company is 'already known' to them. Sometimes the threat of contacting TS can get things done.