Can a pharmacist add a diagnosis to a prescription once the Doctor has already signed it?

No, a pharmacist is not licensed diagnose. Pharmacists only fill prescriptions, though they generally have a firm knowledge base in the field and could make off-the-record, non-affective diagnoses.

A pharmacologist, however, does have the authority determine diagnosis and prescribe accordingly, but generally, a doctor will send a patient directly to a pharmacologist if said doctor does not want to prescribe or is unable to prescribe in confidence an appropriate medication.

That being said, a pharmacist may add a diagnosis if the doctor tells them what that diagnosis is. Some insurance companies require a diagnosis code, which the pharmacist can obtain by contacting the doctor; they may then write this on the prescription for insurance purposes.