Puppies are on the thin side . However, I would encourage your little guy to eat. I would experiment. I belive the vet is concerned because if he get sick then he is not going to have enough reserves in his body to fight off illness.
Have you tried supplementing a high quality puppy canned food? There are YT members on here who know very good people foods for Yorkies. Not greasy food though. That is bad for dogs.
I mixed white chicken meat in with canned food for Suzi, she ate pretty well. Your pup does not like chicken? w/o the skin it can be given. i believe cheese, cottage cheese is ok to give, but, not too much. Keep trying to find foods that he enjoys.
Little Yorkies are known to be picky eaters.
He sounds so adorable
I am assuming he is at least four months? He is going to be just a little dog full grown. My Yorkie pup is nearly 4 months old and weighs 4.12 oz . Sizes vary considerably.