The most common and most likely answer to this crossword clue is the 4 letter word WAFT
I've seen this crossword clue in Independent
W
A
F
T
Buddy explains!

The straight part
We know that "Rush of air" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case rush of air = waft.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does WAFT mean?
WAFT noun- A light breeze.
- Something (such as an odor or scent like a perfume) that is carried through the air.
- A flag used to indicate wind direction or, with a knot tied in the center, as a signal; a waif, a wheft.
- To (cause to) float easily or gently through the air.
- To be moved, or to pass, on a buoyant medium; to float.
- To give notice to by waving something; to wave the hand to; to beckon.