Angular Cheilitis: This condition is characterized by inflammation of the corners of the mouth, often due to fungal or bacterial infections, and is commonly seen in individuals with poor oral hygiene, nutritional deficiencies, or those who wear dentures. It's the most likely diagnosis due to its prevalence and the specific location of the sore.

Cold Sore (Herpes Simplex): Although more commonly found on the lips, cold sores can occasionally appear at the corner of the mouth, especially if the individual has a history of herpes simplex virus infections.
Dry Mouth (Xerostomia): This condition can lead to irritation and sores in the mouth, including the corners, due to reduced saliva production which can be caused by various factors including medication side effects, radiation therapy, or systemic diseases.
Irritation from Dentures or Orthodontic Appliances: Poorly fitting dentures or orthodontic appliances can cause friction and lead to sores in the corners of the mouth.

Oral Cancer: Although less common, a sore in the corner of the mouth that does not heal can be a sign of oral cancer. It's crucial to rule out this diagnosis, especially in individuals with risk factors such as tobacco use, alcohol consumption, or a history of radiation exposure.
Lupus or Other Autoimmune Disorders: These conditions can cause oral ulcers and sores, including in the corners of the mouth, and are important to consider, especially if the patient has other systemic symptoms.

Sarcoidosis: This autoimmune disease can cause granulomatous inflammation in various parts of the body, including the mouth, leading to sores or ulcers in unusual locations such as the corners of the mouth.
Melkersson-Rosenthal Syndrome: A rare neurological disorder characterized by recurrent, often unilateral, facial paralysis or weakness, swelling of the face and lips (usually the upper lip), and the development of folds and furrows in the tongue (fissured tongue), which can also include sores in the corners of the mouth as part of its symptomatology.