What word is there in English for sucking with your nose? There are people who do not sneeze but instead have a bad habit of sucking with their nose snot in again. However, the English dictionary definition for sucking refers to the mouth.
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
Insufflation (Latin insufflatio "blowing on" or "into") is the practice of inhaling a substance.
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'.