Although Prednisone is a quick fix, it's not suitable for long term treatment of allergies.
You should consider getting Benedryl gel caps. Give puppy 50 mg (e.g. 2-25 mg gel caps) at night, and another 50 mg in the morning. You could even give another 25 mg at noon if puppy is still itching. If this gets the itching under control then cut the morning dose back to 25 mg, and eliminate the noon dose. If it doesn't get the itching under control, then ask the Vet to prescribe Hydroxyzine ( a long acting anti-histamine).
The other thing you should be concerned about are the raw spots, as they can frequently become infected with bacteria. Besides the inflammation from the infection, the bacteria proteins can also trigger an allergic reaction. Therefore you definitely need an oral antibiotic. Also a Cortisteroid/ Antibiotic salve will help.
In summary your strategy should be to soak up the histamine, and treat any skin infections with Antibiotics both oral and topical.