Your GP can only provide a private prescription for medication that is not available on the NHS.
A private prescription is not written on an official NHS prescription and so is not paid for by the NHS.
The cost of a private prescription is met wholly by the patient and is dictated by the cost of the medicine plus the pharmacists charge for supplying it.
A prescription is a legal document for which the doctor, who has issued and signed it, is responsible. A doctor you see privately can’t issue an NHS prescription