Suck with a nose

Not to steal your thunder joseph_morris, but while the act is a sniffle as a verb it would be to sniff or snort.

Edit-To expound slightly more, I found this page for UK English slang, but it has a pretty expansive breakdown of snuffling.

The word you're looking for is "sniffle."

Or try the word "snuffle": : to snuff especially audibly and repeatedly

http://www.merriam-webster.com/dictionary/snuffle

Oxford Dictionaries gives as the meaning of 'sniff': 'draw up air audibly through the nose to detect a smell, to stop it running, or to express contempt'.

There is also a verb 'sniffle'meaning: 'sniff slightly or repeatedly, typically because of a cold or fit of crying'.