The most common and most likely answer to this crossword clue is the 4 letter word ASAP
I've seen this crossword clue in The Guardian
A
S
A
P
More clues leading to the result ASAP
- Quickly like a post at the beginning
- Quickly ace goes to moron …
- Son stops drivers parking right now
- Source of apple juice with high priority
- Right away, a step in the wrong direction
- Snake swallows ant’s head very quickly
- Like a P in PDQ
- Like a quiet request to get moving
- Reptile protects its own head at the earliest opportunity
- Has a problem scoring right off the bat
Buddy explains!

The straight part
We know that "Quickly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case quickly = asap.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 ASAP mean?
ASAP adverb- As soon as is possible; at one's earliest convenience; at the earliest possible time.