In order to diagnose the cause of the cough I would talk to your vet about a CBC, endoscopy (after exercise), a tracheal wash and TTA, or a BAL and lung function tests. A combination of these should help get an answer to the cause of the problem. The major causes of horse cough are viral infection such as and equine herpes and influenza, bacterial infections and SAID/IAD which is basically inflammation of the lower airway.

Treatment for SAID is fairly straight forward with initial relief being achieved with a bronchodilator followed by treatment with anti-inflammatories. Bacterial infections can also be solved relatively easily, if it is a viral infection then there is little effective treatment and supplements to improve the immune system are really your only options.