The most common and most likely answer to this crossword clue is the 10 letter word FAST-TRACK
I've seen this crossword clue in The Times
F
A
S
T
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T
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More clues leading to the result FAST-TRACK
Buddy explains!

The straight part
We know that "Urgently promote" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case urgently promote = fast-track.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 FAST-TRACK mean?
FAST-TRACK noun- A racetrack with optimum conditions for high speeds
- A railroad for express trains.
- The quickest or most direct method or path.
- A high-pressure or intensely competitive situation, particularly one characterised by rapid advancement.
- To progress something with unusual rapidity.