Talk to Pediatrician on Persistent Cough Asthmaa Type

Assalaam u alaykum.my son dob 06 march 2014 has persistent cough.we were giving him brittanyl half teaspoon twice a day but during last week he was doing abnormal cough like something irritated his throat or like saliva used to get stuck in tonsils not sure but he used to do cough repeatedly after every few seconds whole day.So we gave him Steron one teaspoon twice a day for just 2 days.With steron his frequentcy of cough decreased to some extent but still he is doing cough a lot.We consider him to be slight asthmatic child having seasonal allergies as well.he also has significantly enlarged adenoids and tonsils for last 3 years or more.His sleep has become much improved Alhamdu Alhamdulillah.Kindly advise which medicine or set of medicines shall we give him regularly for long term during whole winter and spring pollen seasons.jazak Allah ul khayr...child is Alhamdulillah overall healthy has no fever.height and weight are not very low looks a bit skinny though...