My reply from Ayrshire Archives.....
"Thank you for your recent enquiry with regards to finding the burial place of your Great Grandfather, Alexander Gordon who died at Ayr County Hospital in 1893. Unfortunately, although we do hold records of this hospital, there are gaps within the collection and I’m afraid that the Register of Admissions and Discharges for the period 1892 – 1904 has not survived. There appear to be no death registers either, whether they existed and haven’t survived or such a thing was never recorded I can’t say.
We do not hold any other burial records, these are all held by our colleagues within South Ayrshire Council’s Cemeteries Office whom I note you have already been in touch with.
We do hold the Registers of Corrected Entries which were compiled by Registrars and contain information about suspicious or accidental deaths, I have looked for the entry for Alexander and whilst I did find the entry there is no information with regards to his place of burial. The death registration took place at Ayr and the Procurator Fiscal who carried out the investigation into the death is noted as being Stranraer. The other details given are merely his name place and time of the accident and then his subsequent death."
So I think I am at a dead end now [haha]. Seriously though, quite sad I cannot find him.