Question: > 
> **[​sputum](https://www.oxfordlearnersdictionaries.com/definition/english/sputum?q=sputum)** (uncount): liquid from the throat or lungs, especially when it is coughed up (= forced up from the lungs, etc) because of disease
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> blood in the sputum
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> ​[phlegm](https://www.oxfordlearnersdictionaries.com/definition/english/phlegm?q=phlegm) (uncount): the thick substance that forms in the nose and
>  throat, especially when you have a cold
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> ​[mucus](https://www.oxfordlearnersdictionaries.com/definition/english/mucus?q=mucus) (uncount): a thick liquid that is produced in parts of the body, such as the
>  nose, by a mucous membrane
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> a handkerchief stained with blood and mucus
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> The frog’s eggs are surrounded by a coating of jelly-like mucus.
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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**"?

Answer: 
"Sputum" is a fairly uncommon word for most English speakers without medical backgrounds. "Mucus" is a commonly used catch-all, but "phlegm" will also be understood in most cases if you wish to be technically accurate (although I can tell you that the difference between the latter two is unlikely to be acknowledged in a non-medical setting).

If you **are** talking to a doctor, he or she will certainly know what you mean if you refer to sputum, but otherwise you may wind up having to explain your word choice by using one of the other two words.
"sputum", "phlegm", and "mucus" are in increasing order of commonality. I wouldn't expect most people to be familiar with "sputum", but I would expect them to be familiar with "phlegm". 

> 
> Should I say "My child coughed sputum / phlegm / mucus up in general everyday English"?
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I take it you mean "Should I say 'My child coughed sputum / phlegm / mucus up' in general everyday English?"