There may not be a word for it, but maybe there is a better or more concise way of describing it.
Snuffle. v.i. To breathe noisily through your nose, for example because you are crying or have a cold.
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I would say hawk - to raise by trying to clear the throat (hawk up phlegm).
Some will say that - as with snort - the movement of mucus is up/out rather than down/in. But often it's a composite move - down the sinuses to the back of the throat, up into the mouth, then out (spitting).