Looking for a fancy way to say "cough up phlegm"? Try expectorate.
There's no way around it — expectorating is pretty gross. If you have a respiratory problem like bronchitis, you're going to expectorate a lot: phlegm and mucus will build up in your lungs, and you'll eventually cough it up and spit it out. A doctor might prescribe a special cough medicine to help you expectorate. Sometimes people use expectorate to mean spitting out plain old saliva, too.