Why won't my cat's spreading neck sores heal despite vet visits?

Those are some very nasty looking sores. I can only imagine how uncomfortable they are for Gypsy. I also notice in the picture that she is not wearing any sort of cone around her head to prevent her from trying to lick the affected area. Even if it in a place where you would think they can't reach, they usually find a way. I would also consider placing little booties on her feet to prevent scratching of the area. In terms of the actual problem, if your veterinarian has already tried multiple treatment without success, it may be time to start trying some diagnostics instead. These tend to be a little more expensive and time consuming, but ultimately will get you the answer that you have been looking for all along. Some of these tests may include scrapings, cultures, skin biopsies, and allergy testing. Alternatively, you could seek out the help of veterinary dermatologist who specializes in these types of issues.