Hi, thanks for the query to H.C.M.
is a life saving drug and it should uses in only emergencies.
Just for mild illness it's use in an unethical thing.
Cough with expectoration is an indication of infection with allergy
Productive cough is an indication of Bronchitis
or lung abscess
In my opinion there is asthmatic allergic bronchitis (AAB) causes are mite, dust fine particles, powder, pollen etc i.e. inhaled foreign bodies are main reason.
Pl be sure that prednisolone , no way, keep it only for an emergency cases.
Get in blood examination and x ray chest to rule out any disease.
Get in examination of sputum.
Regards. Dr. HET