Many dogs have nose problems if they have immune disorders. This is pretty common. Because of this, you need to protect the nose the best way that you can. This means applying sunscreen every day. At night, you can use a moisturizer called "bag balm" (walmart) and coat the nose liberally with it. Some pets with this disorder will actually get so bad that the vet suggests tattooing the nose to prevent the peeling and scarring.
If an allergy is causing the problems :
I would talk with the vet about a different low allergy diet, one that will have the least reaction. Fish and potato works well as does rabbit and potato. Most dogs with allergies do well with a diet that does not contain corn or wheat.
You should also get her on a flea preventative if she is not already. Even if you don't see fleas, it only takes one to get the cycle started. Some dogs are highly allergic to flea bites.
I would also use a prescribed dose of benadryl, especially during allergy season. My vet recommends 1 mg per lb of body weight given every 8-12 hours.
Keep in mind that dogs can have allergies to pollen, dust and other household allergens so you should evaluate your home and try to make it as allergy free as you possibly can for her.
It sounds like the dog might also have a secondary bacterial or fungal infection. This is usually secondary to allergies and itching. This means that the secondary problem is caused by the primary.
When dogs scratch, they abrade the skin and allow bacterial and funguses to enter the skin. The result is a bacterial or fungal infection.
You need to see the vet so that they can get a culture and tell you what kind of treatment is necessary.
While you are there, you should also speak with the vet about a supplement that has omega three fatty acids. Good skin starts from the inside out and this can be very beneficial.
Let me know if you still have specific questions about this.