Thank you for your query,
lasting for 1 month is uncommon.
I think this may be due to repeated upper respiratory tract
infection in a month.
Usually antibiotic are given during infection for 5 to 7 days and this duration is enough to treat routine upper respiratory tract infection.
But cough requires only symptomati treatment with cough syrups depending on the nature of cough.
, and expectorants may be prescribed depending on the type of cough and some times anihistamines are required in allergic cough
So I advise you to revist your Pediatrician
for the examination to look for the causes of cough and the appropriate cough syrup.
Hope I have answered your query.