Three years ago, I had tenacious phlegm that I'd have to work very hard at to hack up and spit out. After a lot of tests, including allergy testing, scoping, MRI and X-rays, nothing was found, but it went away. Although I haven't smoked for 40 years, it's now returned and I need help to figure out how to be able to get the phlegm out. Coughing is a complicated but important reflex to protect the lungs and airways against foreign particles. Any irritation of the airways and bronchioles in the lungs will likely cause excess secretion of bronchial mucus.
Sputum is a mixture of the foreign particles, mucus and cells expelled by the lungs.
An unusually large amount of sputum is also called phlegm, which is a sticky, viscous secretion produced by the mucosa of the respiratory passages and discharged through the mouth. You are right to try to bring up the phlegm to clear your airways. Coughing that brings up a lot of phlegm probably should not be suppressed. While I appreciate your concern and interest in finding a way to clear the phlegm, the best course is to keep it from forming in the first place. So the first step is to identify the source of the problem.