I have asthma and COPD what could cause my lower part of my left lung to hurt when I cough?

Neither. However if you have had some violent coughing spells or severe asthma attacks, you might have sprained some of the muscles on the rib cage. Inflammation of the lung lining at that site may also cause pain when coughing. If NSAIDs does not help or the pain gets worse, you would need to see your doctor.