My son cough a lot before when he was 3 years old and my doctor gave him 2 drugs that took by inhaler. The doctor said the drugs can go directly to his lung. Although it is expensive, about $700 or so for the inhaler (it is bigger than adults) and the medicine but it is worth it. He recovered in 5 days.
Dr Barbara Lam in Central is very good - my son has ongoing problems and she has been excellent. i beleive there is another doc in central who is good but ican't remember his name found it on a search once.