Unlike humans, only about 30% of dogs respond to anti-histamines so we usually have to resort to stronger medications to relieve allergies. Oatmeal shampoo is very soothing and antibiotics are often necessary to treat any infections that arise. Skin allergies are extremely frustrating for everyone involved and can be very debilitating in badly affected dogs.
The problem with skin irritation or dermatitis is that there are many different causes that can present with the same type of rash and distribution. Some of the common causes of itchy skin and rashes are flea allergy, contact allergy, food allergy, mange and atopy. The problem is that without a methodical, diligent trial to rule out each possibility one by one, you will probably never work out the cause!
I would suggest talking to your vet and explaining that you’d like to get to the bottom of the problem and conduct some trials (if they haven’t already been done) to work through the possible causes. You could also ask about omega fatty acid supplements to improve the quality of your puppy’s coat. Anti-inflammatories are fantastic at treating the itch but long term use is usually avoided. Fortunately in recent years new drugs have become available to treat skin disease. In cases of atopy (usually diagnosed when all other differentials have been ruled out) patients can be allergy tested and a personalised vaccine administered which can desensitise their body to the offending allergens.
Your veterinarian will be the best person to advise you on the most appropriate next step for your puppy.