Hello and welcome to the forum Jonathan. I am not familiar with Lily's Kitchen food so I have just checked on the Dog Directory and found the puppy dry food review - link.
It has a nutritional rating of 80% and the ingredient list is quite lengthy.
At 8 months old, I would be inclined to discontinue puppy food. The main difference between this and adult food is the percentage of protein. Puppy foods usually have a protein content of circa 30% and this particular one is 31.5. Many high quality dog foods have an equivalent percentage, some being higher so it would be ok if you look for a generic product in this category rather than a puppy one.
There are so many good quality dog foods that I cannot recommend a particular one but you can search using the Dog Food Directory
on this website. I usually use the filters clearly labelled and no red/yellow ingredients just as a start. If you turn the ratings slider up to 80% to 100% it will return the better quality products.
Overfeeding can cause the problems you describe so I need to ask - are you weighing the food accurately? Is anyone in the family feeding extras?
At eight months old it is unlikely that your dog has developed an intolerance to ingredients but it would be worth going for a product that has a fairly simple recipe. One group of foods that are worth looking at are the cold pressed types. They are quite digestible and generic so you would not have to move your dog onto another product as it gets older. These can be found using the Dog Food Directory and ticking the relevant box under Type of Food.