Why is my dog losing fur and having discharge under its eyes?

From the photo you uploaded, the dog may have conjunctivitis, which causes more eye discharge that soils the fur under the eyes, leading to discoloration and hair loss. For this, the owner must clean the dog’s eyes, wipe the discharge promptly, and keep the area around the eyes clean. Then use a pet-specific anti-inflammatory, antibacterial eye drop such as chloramphenicol or ofloxacin to reduce the eye inflammation. If the dog does not improve after medication, stop the drops and take it to a veterinary hospital.