The most common and most likely answer to this crossword clue is the 5 letter word HASTE
I've seen this crossword clue in The Times
H
A
S
T
E
More clues leading to the result HASTE
- Hospital teas prepared in great speed
- Holds note, displaying urgency
- Leader abandons modest expedition
- Speed in heats, uncommon
- Eats meal briefly, in rush
- Be unable to stand, barged into by son in hurry
- Really dislike admitting son’s inconsiderate speed
- Catholic leaving continent with undue speed
- Speed, heroin & ecstasy going round varsity every now and again
- Start off continent expedition
Buddy explains!

The straight part
We know that "rush" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case rush = haste.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 HASTE mean?
HASTE noun- Speed; swiftness; dispatch.
- Urgency; sudden excitement of feeling or passion; precipitance; vehemence.
- To urge onward; to hasten.
- To move with haste.