A lot of people, especially those with a throat infection, make an oft irritating guttural noise to clear their throat. What is the act called?
Historically, hawk (also hock) meant to clear the throat in the general sense. Now it refers to gathering phlegm for spitting, by a progressive closing of the tongue against the back of the throat, accompanied by an outrush of air, a rising pitch, and vocalisation.
Retch also meant to clear the throat or cough up, from the Old English verb hræcan, to cough up or spit phlegm (hraca). Now it refers to the sound and motion of vomiting, either gagging or dry-heaving, often with a falling guttural sound.
So while we have had two good strong English one-syllable verbs to express what you mean, they’ve both drifted into new roles. In the present, I think ahem is the best verb we’ve got:
My coworker Bill is just getting over a throat infection and was ahemming all day. Better than last week, when he would hawk loogies into his handkerchief.
It sounds a little 19th-century, but I do like to harrumph. It's the sound of clearing one's throat, although not necessarily to gather mucus.