At the opticians, such as Boots, they use photographic technology to capture an image of your eye. Given that the image in reality is my property as it's my eye they're taking a photo of - can I request access to the photo but also not permit any distribution of it, i.e. can I ask them to send me the photo and they having to delete it right after?
Copyright in a photo belongs to the photographer, not to the subject. A picture of S (the subject) is not S's property in any sense, unless the photographer has given or sold it to S.
If the photo was linked to information about S, it might be personal Data under the GDPR (or UK-GDPR) if so, S could request erasure, but various exceptions might apply. But even then S would not aquire the right to distribute or copy the image, or any of the other parts of copyright.
David Siegel's answer is the correct one from a legal point of view, but I wanted to add some additional information:
The software that their machine uses is likely to store the image in a specialist format. GDPR requires portability of data. This means that if you request the image then they are required to give it to you in a format which can be imported by a different optician to use it for similar medical purposes. I do not think it means that they would have to convert it to a general purpose image format that you could use (for example) to print it out and put it on the wall. They are not set up for this and will not know how to do it, even though it is technically possible.
As an example, I once asked at the dentist for a copy of my x-ray, because I was at an out-of-hours place. The dentist promised he would send the image to my usual dentist. He didn't have a clue how to export the file from their system, but he also let me take a photograph of the x-ray on his computer screen using my phone.
When I got to my normal dentist he said he had not received information from the other place and so I would have to have another x-ray. I showed him the picture on my phone and he was happy that it was good enough to see what he needed to, so in this way I saved myself from an unnecessary dose of radiation.
I don't think I had the right insist on taking a photo of the x-ray, because they are allowed to take up 30 days to respond to GDPR requests, but in this case they were willing to do the simplest thing, which served my purpose.