I would be 95% sure it is the RC.
The same thing happened to Ellie.
She was on it for less than a week and started scratching like mad.
She got a steroid injection and the food was taken away.
She has had food allergy issues ever since.
Unless there is a medical reason to feed RC, I just do not recommend it for Yorkies. Way too many get itchy from it. It's easy enough to try it and then stop if it causes a problem, but then you face months of the itchies. I'm not saying it's a bad food but sooooo many people on YT are reporting this problem.
Anyway, the steroids helped some (Benadryl not at all but you could try) and then she got a lot better until the next allergen/food sensitivity trigger cam earound.
It can take 2-3 months for you to see a change. He will likely need drugs before then. You can always go to the vet so they can rule out other issues but it may be a waste of money.