Driving test cancelled half way through because of problems with car, can i claim money off my instructor?

Your driving instructor has a duty to insure that the car they teach you in, complies with road safety. A faulty indicator is a breach of their duty. Therefore, they have to refund the cost of the driving test and the driving lesson took on that day. All you have to do is complain to the driving company or DVLA. The difficult part is, whether you were learning with a fully established driving company like BSM or with an independent.