If you have treated your chronic cough with antibiotics and anti allergics, there is much chance of cough being dried up and settled as it is at its place. And thus is creating pain and discomfort when you need to breathe heavy ex during exercise.
Next Steps
we need to rule out first if its muscular, for that simple stretching exercises of shoulders will do.
Health Tips
if you feel breathe less or pain while breathing heavy after workout you need to do an x ray chest .