Cough syrups simply suppress Jimmy’s urge to do so. This is not necessarily a good thing, since coughing keeps unwanted substances like dust and germs out of the airway. In asthma, especially, the cough helps by bringing up excess mucus. As a rule of thumb, if Jimmy is coughing without complaint, don’t interfere with his expectoration process.
My approach is to prescribe cough syrups sparingly, but when I do prescribe them, I use the ones that really work. The codeine-based syrups are the most effective, and they have relatively minimal side effects.