The most common and most likely answer to this crossword clue is the 4 letter word FLEW
I've seen this crossword clue in Independent
F
L
E
W
More clues leading to the result FLEW
- Went on air and broadcast complaint
- Illness reportedly went quickly
- Infection of ear
- By the sound of it, viral disease went quickly
- On the radio, channel ran well
- Only one or two taking Latin passed quickly
- From what we hear, disease was airborne
- Not many will cross line winged
- First two sections of florid central news took wing
- According to report virus was airborne
Buddy explains!

The straight part
We know that "went quickly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case went quickly = flew.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 FLEW mean?
FLEW noun- (chiefly plural) The thick, dangling upper lip of certain breeds of dog, or the canine equivalent of the upper lip.
- To travel through the air, another gas or a vacuum, without being in contact with a grounded surface.
- To flee, to escape (from).
- To cause to fly (travel or float in the air): to transport via air or the like.
- (of a proposal, project or idea) To be accepted, come about or work out.
- To travel very fast, hasten.
- To move suddenly, or with violence; to do an act suddenly or swiftly.
- To display (a flag) on a flagpole.
- To hunt with a hawk.
- Shallow; flat