Hello Yorkshiregirl. I have merged your thread with the previous one as it is the same subject. I have looked at Barking Heads Puppy food and it attracts a good nutritional rating - link.
There are a number of posts about allergy and you can find them by using the search box at the top of the board index.
Also, it might help to have a look at this video: Alison Daniel from My Pet Nutritionist talking all things allergy.
Finding the cause of itching/scratching can be difficult. We automatically assume it is food but it can be due to environmental factors or, as you have mentioned, genetic factors. Have you seen your vet about this? Is your puppy insured? If you can find a vet that is particularly interested and knowledgeable about skin problems it can be extremely helpful and save money in the long run.
Skin scrapings are useful and you can ask your vet about this. There are two common issues that arise with intense itching/scratching - bacterial skin infection and yeast infection (malassezia). These can be treated but very often need intensive therapy.
My feeling is that it is best not to feed dry extruded food to dogs with this type of problem. An elimination diet can pay dividends but most vets would start the dog off on hydrolysed protein dry food. Most dry food is high in carbohydrate and with an itchy dog you want to look for low carb.
Quality wet food with a very simple recipe (single protein) might help. It is said that beef and chicken are common allergens but I have found that white potato and legumes can also be problematic. It’s not uncommon for dogs to improve on a raw diet as this is usually low carb. If you are inclined to try this use a good quality complete product. The other option is fresh cooked food.
Have a think about this and let us know what would be acceptable.