The most common and most likely answer to this crossword clue is the 5 letter word SCOOT
I've seen this crossword clue in The Daily Telegraph
S
C
O
O
T
More clues leading to the result SCOOT
Buddy explains!

The straight part
We know that "Quickly leave" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case quickly leave = scoot.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 SCOOT mean?
SCOOT noun- A dollar.
- A scooter.
- A sideways shuffling or sliding motion.
- To walk fast; to go quickly; to run away hastily.
- To ride on a scooter.
- (of an animal) To move with the forelegs while sitting, so that the floor rubs against its rear end.
- To move sideways (especially along a seat for multiple people), usually to make room for someone else (to sit, stand, etc.).
- To dispatch someone or something at speed.
- A sudden flow of water; a squirt.
- To squirt.