Thanks for posting your query at DoctorSpring.com.
I can understand your concern. I reviewed the picture.
This is a furuncle which is a bacterial infection of the hair follicle. When many furuncles combine its called a carbuncle.
You would need to take an oral antibiotic like azithromycin. Just with application of neosporin the lesion is not going to heal.
Azithromycin requires a prescription.
I understand that you do not want to see doctors but you will need a prescription without which the lesion won't heal.
Please let me know if you have any queries. Take care