I don't know about eating it but it's really good as a wound dressing. Just apply liberally to cyst and then apply clean dressing. As other people have said the antibiotics are only part of the solution, good sound management are what will probably solve the problem, ie having the cysts lanced, regular cleaning and dressings. As I've said before, skin infections are the secondary problem and you probably already had the bacteria on your skin. It's the different antibiotics that have made it into a super bug that might be resistant.
I would imagine the antibiotics help to get rid of some of the infection but the underlying issue is not resolving, hence why it's not clearing up completely. Then the bacteria comes back each time becoming more resistant as it's the ones that wernt killed off by the previous antibiotics. The doctor def needs to do a culture and sensitivity but if I were you I would be wanting to see a dermatologist as it doesn't seem like the gp is addressing the real issue.