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.
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).
Where I live, it is mostly called a sniffle (close to snuffle I know, but not the same.) A person who has a mild head cold that causes them to do this is said to "have the sniffles". Presumably this came from the word "sniff", as that is basically a special case of sniffing (one where taking in an odor isn't the main purpose).
I have heard "snuffle" before, but had always assumed it was an alternative from people who didn't hear the word "sniffle" right, and were just going with it. Given the popularity of the other answers here with it though, I suppose there might be dialects where it is more common. It certainly isn't in the USA South Midlands area I live in though.