Talk to Pulmonologist / Lung on Syrrup For Cough

assalamu alaykum. my child age 5 years 9 months has pollen season starts or winter dry cold seasons starts he has continuous hes inherited allergies from me and my wife also has cough evrry winter. With start of dry cold winter We started giving brittanyl one tea spoon twice a day(but brittanyl hasnt shown significant improvement).Now started giving myteka 5 g chewable once a day from put 1 drop of Betnesol N nasal drop twice a day per nostril. Shall we change is full of phlegm.

Phlegm comes out with cough but since child is below 6 he cant properly spit phlegm like he throws phlegm becomes severe and he he gets relief for few has been diagnosed as having mild asthma and enlarged adenoids and tonsills.

Please advise medicines for cough that can clear his chest of phlegm. Also please advise regarding medicines already being given??? Alhamdu lillah he has no