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No one online can tell you to give prednisone to your dog as it is a prescription drug. However, some experts do believe that prednisone can in fact trigger demodectic mange so since the dog already has mange, it woud not be advised to give prednisone at this point. The ivermectin often clears up demodectic mange pretty quickly though it is likely to take months since the demodex is generalized.
Now what you might try is Shampoo containing Chlorhexiderm and/or Oatmeal which helps soothe the skin. It can also help with secondary infections which is what you want to help prevent. Another product that I've found seems to help with the itchiness is Happy Jack Mange treatment, but I will warn you that it smells like sulfur and stains everything it touches. It is not a good idea to have your dog in the house with this on it. However, I did see hair regrowth in a week with this product but I was also using ivermectin as well.
One thing that might help with the scratching is getting an infant t-shirt or onesie. You can cut the shirt under the armholes and put it on your little dog. If he scratches, the nails will be less likely to break the skin. You can also be sure to trim his nails so they are not as sharp either.
I hope this information is helpful to you. If you would like any additional information or have more questions please don’t hesitate to press the reply to expert or continue conversation button so I can address any issues you still have . If you do find this helpful, please take this opportunity to rate my answer positively so I am compensated for my time.