Hello n welcum to hcm, the most important thing here in your kid's case is recurrency of cough despite of allopathic medications. It shows that your son is having very weak immune system.
Please do not ignore this recurrent and worsened cough as it can be the symptom of underlying disease such as pnemonian, bronchial complaints, pnemothorax, childhood asthma, pulmonary congestion etc. Please do not ignore the thing.
Else you can go for the complete hemogram, sputum analysis, xrays of chest also to clearify the doubts. Post diagnosis please consult a good physician there in your area for further prognostics. It us for sure that your son is ving very low immune response so you can consult a good homoeopathic physician also for immunity enhancement..... Take care, god bless...