Do not change food, if it works well for your cat's age, weight, and breed and is vet recommended. Keep the new food if everything else is fine and healthy. Cats usually don’t like change in their food.
There are supplements like collagen that you can add to your cat's food that may help her coat and be more cost-effective than switching to an expensive food ...
According to my thinking, this is a bacterial skin infection specifically known as the staph infection caused by staphylococcus aureus (bacteria). It usually doesn't cause irritation, itching, or swelling. But in severe cases may cause redness, inflammation, and bald spots on the skin.
I think visiting a veterinarian clinic would be a great idea.