sniffle

To make a whimpering or sniffing sound when breathing, because of a runny nose.

Her dad heard her sniffling and passed her a handkerchief.

To utter with a whimpering or sniffing sound.

He sniffled a pathetic apology.

The act, or the sound of sniffling; the condition of having a runny or wet nose, as from a cold or allergies.

I sometimes have a sniffle during allergy season.