Thank you. As you will be aware, the position here is that you have two differing sets of rights. The first set of rights you have under any warranty from the dealer which are somewhat limited insofar as that they are limited to the terms of the warranty which are probably not overly generous if indeed any warranty was offered at all or the warranty has not yet expired as I think is the case here. I will leave this here though because of what follows.
You have another set of rights by virtue of your contract with the dealer from whom you purchased the car. This set of rights are much more extensive as terms are implied into your contract by virtue of the Consumer Rights Act that the car must be of satisfactory quality, fit for purpose and as described. During the first six months, any faults are automatically assumed to be inherent as opposed to having been caused by damage - in other words the dealer has to prove they were not present when you bought the car and have been caused by damage by you; you do not have to prove that you have not damaged the car. After the first six months, the burden of proof switches to you to show that any faults are inherent as opposed to having been caused by damage though this is not usually difficult as damage is usually fairly clear - e.g. physical damage by an accident failure to maintain properly such as keeping oil topped up and so on. You may need to obtain a mechanics report from an independent garage as to the cause of the issue as evidence which I think from what you say you may already have.
The Consumer Rights Act gives you the right to reject a car if it is faulty but that right only exists for a short period - a little under a month, but you still have rights under the above legislation. Under these rights, providing you can show the fault was caused by a fault and not damage caused by you, you can insist upon a repair or replacement at the dealer's discretion. You are required to give the dealer reasonable opportunity to repair at least once. If the dealer cannot or refuses to replace or repair the car or fails to repair the car more than once, then you can seek as an alternative a price reduction or exercise a final right to reject the vehicle and return it for a full refund less any reasonable deduction to take account of your use of the car.
In terms of how you may consider proceeding. You may wish to write to the original dealer inviting them to consider repairing or replacing the car on the grounds that it is faulty and the faults existed at the time you bought the car and were not caused by you. If they do not respond or you are not satisfied with their response then you can consider writing to them again advising that if they do not reply with their substantive proposals within 14 days of the date of your letter you will have little option but to issue proceedings against them in the county court.
If you decide to issue proceedings for breach of contract as above the simplest way to do this is by using the money claim online court service.