The most common and most likely answer to this crossword clue is the 4 letter word URGE
I've seen this crossword clue in The Times
U
R
G
E
More clues leading to the result URGE
- Press crumpled rug with energy
- Some honour geisha's desire
- Withholding power clear out press
- Drive ultimately ending in river
- Encourage heading away from sudden rise in water
- Fancy liqueur gentleman bottles
- Longing to remove initiator of political clean-up
- Head going off to purge press
- Desire that’s in our genes
- Impulse our gentlemen must restrain
Buddy explains!

The straight part
We know that "press" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case press = urge.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 URGE mean?
URGE noun- A strong desire; an itch to do something.
- To press; to push; to drive; to impel; to force onward.
- To press the mind or will of; to ply with motives, arguments, persuasion, or importunity.
- To provoke; to exasperate.
- To press hard upon; to follow closely.
- To present in an urgent manner; to insist upon.
- To treat with forcible means; to take severe or violent measures with.
- To press onward or forward.
- To be pressing in argument; to insist; to persist.