sputum (uncount): liquid from the throat or lungs, especially when it is coughed up (= forced up from the lungs, etc) because of disease
blood in the sputum
phlegm (uncount): the thick substance that forms in the nose and throat, especially when you have a cold
mucus (uncount): a thick liquid that is produced in parts of the body, such as the nose, by a mucous membrane
a handkerchief stained with blood and mucus
The frog’s eggs are surrounded by a coating of jelly-like mucus.
My child has a mild sore throat. Sometimes, she coughs up some thick sticky gel-form liquid (it could be white if it is a mild illness, could be more yellowish if the illness is more severe)
Ac coding to my research, "phlegm" & "sputum" are the same and imply when you have a cold / throat infection, etc, but not sure which is more general common term. But "mucus" does not imply a cold / throat infection, etc. However, I heard a lot of native speakers use "mucus" even for a cold / throat infection.
Should I say "My child coughed sputum / phlegm / mucus up in general everyday English"?