There are various opinions on what food is best...there really is no set answer to that. However, if the pup has unknown allergies as well as coat/hair issues, I would start with switching to a grain free food. As I said, there are different ones but I will suggest what I know. Mandie eats Blue Buffalo Wilderness. I get her the small breed but they have a toy breed variety that has even smaller pieces. If Roscoe Liam likes the smaller kibble, I would try that.
You may never know what his allergies are if you are on a limited budget, but you can test things out. Try grain free food first...that is the most practical starting point. After a bit, if that isn't changing the allergic symptoms, talk to your vet about trying proper dosages of benadryl. Wipe paws with wipes when coming in from outside...may be a grass allergy. Keep treatment for fleas current to keep any flea allergy at bay.
Hope this helps some....just tossing out ideas.
