What things can cause moderate, visible mucus on or in stools?

The most common are infections due to bacteria, parasites or fungi, however it can be due to inflammation of autoimmune or allergic origin. You should go to your physician and check out whats the source

There are many causes mostly benign and transient as mucus is part of normal stool. But if excessive one might have to consider, bacterial infection, Inflammatory Bowel disorders like Crohn's Disease. Even Intolerance to lactose, fructose, sucrose or gluten. While probiotics and healthy fibers in diet can help, best to see a Gastroenterologist if symptoms persist.