I can think of several anime series with recurring characters who I would classify as "snot-nosed kids". Two sympathetic, recurring, characters are Udon from Naruto and Komugi from Hunter x Hunter. Is there an actual trope for this character trait, and if so, what is it meant to imply? To me (American), the trait merely means someone I don't really want to look at.
Surely there is more than that, though. Sort of like a bloody nose often means the character is somehow sexually embarrassed.