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The burn injury you have will take time to heal. You can help it along best by putting nothing on it and sitting in water with epsom salts twice daily. That will be soothing and healing.
The redness and swelling may mean that the burned area is infected. If that's true, you'll need antibiotics, and neosporin won't really help, because the infection is under the surface.
I would use the epson salt baths, but I would also see a doctor for evaluation. The doctor will hopefully be able to tell whether antibiotics are necessary, and can give you a prescription as appropriate.
Fro the itching, I would take benadryl 25mg 1-2 3 times daily. If you need to, put a little 1% hydrocortisone on it as well. Best wishes.
If there is no infection and only burn injury, it should be feeling better within the next couple of days.
However, infection delays healing. If there is infection under the skin, then it won't get better until the infection is resolved.
Injured tissue is more susceptible to infection. If any bacteria get under or through the skin, infection can set in.
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