If the faults were all there when you purchased the car then it is really down to the garage to fix the issues for you, free of charge, and not claim off the warranty.
If the garage does decide to claim off the warranty then it would be wise for you to read your warranty documentation thoroughly. I am assumming that it will be an aftermarket insurance warranty.
Of course you have your rights and if the car is not fit for the purpose then you can always return it, however best to take advice on this, as there are procedures to be followed.
Having finance on the car is a real benefit in situations such as these, especially if you have a conditional sale agreement (HP) as the car belongs to the finance company until you make your last payment. If you have taken out a personal bank loan on the car then you are the owner of the car.
Lean heavily on the finance company and the general manager at the garage and you should get a result.
Good luck.
Alex.