Tell me VictoriaX did your vet have you do food trials to determine what your cat was allergic to or did he or she just tell you what they thought your cat was allergic to?
You really can't find out without a food trial which takes weeks and you never said if your vet prescribed an antibiotic for the underlying skin infection caused by the cats' scratching and licking all it's fur away from it's neck an beyond.
I had a cat with this very same issue If you did at least a 6 to 8 week food trial I would believe what your vet said your cats allergies are. If not I highly doubt that cat is allergic to all three ingredients. Plus did the vet treat the skin infection?
Food trials consist of taking a cat off all normal protein sources and feeding them unusual ones until they heal. They will not be allergic to Venison or Duck. Natural Balance canned cat food venison and Duck limited ingredient cat foods are the best and easiest to do this with. It is what I used to cure the allergies along with Ciprofloxacin to cure the skin sores and infection. The you next need to find the real culprit of the allergy. Please Google Food Trials for allergies in cats if you haven't done one yet. You don't want to limit your cat to such a restrictive diet if you don't have to. He or she will be harder to place if you do. I swear it will only take 2 to 3 months max to fix this whole thing. Just switching your cats diet without addressing the bacterial infection under the skin won't work. Plus you need to find the allergic culprit if your vet just told you what the cat was allergic to without doing a food trial. There is no way for the vet to know otherwise without a food trial. They are just guessing without a food trial. Believe me, I have suffered with this issue just as you have and I know what's up. I know what you and this cat are going thru. If you check in here I'll tell you exactly how to cure this cat in the shortest time possible. I can tell you the right antibiotics and doses and the right foods to fix it. Then I'll tell you how to figure out the exact culprit that's causing this all without making the cat sick again.
Below are a few links that'll help from some vets who know about how to treat food allergies. I know my own vet wasn't much help either so I cured my cat on my own with vet guidance and a ton of research and speaking to other vets. I also did it for much less money.
My cat ended up being allergic to any type of fish product or protein. I thought it was chicken he was allergic to also according to my partially uninformed vet. Thankfully that wasn't the case and the way I found out was by doing a food trial which all vets usually recommend when it comes to food allergies.
BTW vets don't learn much about cat nutrition in vet school. Thankfully it's becoming a much more studied topic nowadays. It used to be the big pet food companies would come into the vet schools and teach nutrition to the up and coming vets. What a train wreck that is. Pet food companies up until the last 10 years haven't really cared about our pets best nutrition until the last few years when this issue has come to light. Profits were their main goal for years. The right nutrition can help an animal avoid so many diseases. It's just amazing what a good diet can do. Allergies are no less important.
Dog and Cat Food Allergy Treatment - Treating Canine and Feline Food Allergies
Identifying Food Allergies in Cats with Food Trials - VetInfo
There's a lot more info out there if your interested. You do need to address the infection too though. For me 250 mgs. of Cipro twice a day ground up with a pill grinder and mixed in with the canned food worked wonders during the trial feeding duck, venison or rabbit. Natural Balance has the right type of food to feed during a food trial to cure your cat. They have the Venison and Pea and the Duck and Pea food. You need to feed an unfamiliar protein and carb to start the trial. The antibiotic the vet gave me at first worked initially but as soon as my cat was off it the issue came back. A second round with that antibiotic did nothing. Ciprofloxacin was the key for my cat. It's a very common antibiotic for humans and animals that works.
Good luck and please message me if you want to know more.