Welcome to YT! It sounds like he may have IBD or another digestive disorder. Many of us who have Yorkies with digestive issues have had great luck consulting with Susan Davis in CA (phone consultation). She is a pet nutritionist.
It sounds like some probiotics, digestive enzymes and a specialized diet may be in order. Lamb can be high in fat and difficult to digest. White rice can also be hard ona sensitive tummy. For those with digestive issues, very easy to digest proteins and foods without any grains can be a key in getting it under control. Generally white fish (talapia) is an easy to digest protein.
Here is Susan's website if you'd care to take a look...
Pet Health Pet Nutritionist Holistic Pet