Welcome to Connect @alicoleman84
Reading through this thread it appears that this is a common issue that is not easily resolved. I have not experienced this problem myself, but will share my opinion after 40 years as a veterinarian and 12 years with ENT experience battling Head and Neck cancer.
The commonly prescribed antibiotic Mupirocin is used because the doctors are assuming that the bacteria causing a small localized infection may be susceptible to it. Many of us, even when healthy, carry MRSA in our nasal cavities. It doesn't cause a problem until maybe there is a break in the skin or mucous membrane that allows it to penetrate deeper, creating a chronic infection there. This organism can be susceptible to Mupirocin but needs long term use to conquer it.....like weeks to months.
As a side note, I have had good luck with using medical grade Manuka Honey on animals with chronic skin lesions. It is not only antibacterial but it stimulates healing. Medical grade is different from the honey we buy in the store. I get it via Amazon. and Manuka Honey comes from New Zealand. It can be messy to use but I have seen great results in otherwise unhealing cases.
I would recommend that any chronic sore be checked by your physician, as skin cancers can present as a lesion that will not heal. Have you consulted your PCP or an ENT about this issue?