This is close to unbelievable. If what they are saying is true (it's a common issue with no known fix that takes the car off the road), then the forum would be littered with complaints, and it isn't (yours is the first I've seen). If they really can't fix your car, then they should admit it & replace it so that you can move on. You're paying a lot of money each month to have the use of the car you chose, not some cr**py corsa. You deserve better service than this when you enter into a binding pcp agreement.
Only the legal owner of the car can reject it. If you've bought the car on a pcp then the equivalent course of action would be to sue the pcp provider for breach of contract.
Either would be a massive step involving a great deal of hassle, and potential legal costs which could be more than the value of the car. You'll almost certainly need legal advice if you do this, and that won't be cheap - solicitors typically charge at least £200 per hour, plus VAT. You can get free advice from your local CAB; it's just possible that they will be able to exert some pressure on the relevant parties to get this sorted out, but don't count on it.
I strongly recommend that you deal with this through amicable negotiation if you can. The most important thing to do now is to make the pcp provider aware of the situation, if you have not already done so. This would in any case be a legal requirement before taking formal action, but the emphasis at this stage should be on getting the pcp provider to put pressure on the supplying dealer to sort this out by whatever means are necessary.
Whatever else, don't stop making the monthly payment to the pcp provider or you'll likely have not just a ruined car, but a ruined credit history as well. Fixing that could prove even harder than fixing the car.