The quickest way to resolve things is to go to the Vet, and ask for some Cephalexin, to treat from the inside, and a combination antibiotic-cortisone drying agent to apply on the outside. Also ask the Vet for the appropriate dose of Benedryl to give to your dog (e.g. clear gel caps with peanut butter on them go down fine). Or the Vet may prescribe a longer acting anti-histamine. If the skin is dry and flakey have the dog checked for low thyroid. More likely than not you have an undetected flea, or an environmental allergen in combination with the higher summer humidity triggering the allergy- and not the food.

In the mean time if you want to keep putting the "stuff" on puppy, and not have the other one lick it off, then after applying it, put on a thin film of tea tree oil on the very top of the "stuff". All it takes is one lick and the other puppy will not lick again.

As a supplement after you get the current situation under control try fish oil, and Vitamin-E (dry succinate salt form for best absorption). Ask your Vet for the appropriate Vitamin-E dose to give. For the fish oil you can give 2000-8000 mg per day depending on the size of the dog. Worst case scenario is loose stool if too much is given. Dogs will gobble down fish oil caps with little encouragement needed. The fish oil acts as a mild anti-inflammatory, and the Vitamin-E helps aid in the repair of damaged skin.