A culture is usually done on sputum to identify the pathogen and the sensitivity is done on same specimen to see which antibiotic will kill the organism.

The bronchiectasis can cause a cough because we have a harder time getting the sputum out and the body keeps trying. It's why we do airway clearance. The H. flu can certainly make coughing worse and once treated makes most people better. As Sue said, if you don't improve, call doc so antibiotics can be adjusted. Good luck to you.