The most common and most likely answer to this crossword clue is the 10 letter word ALLEGRETTO
I've seen this crossword clue in The Times
A
L
L
E
G
R
E
T
T
O
More clues leading to the result ALLEGRETTO
- Quickly, singer grabs large wader
- Tricky to greet all in a brisk movement
- Briefly hold mammal spinning round quite briskly
- Rather lively warbler catching lake bird
- Claim docking tail repelled aquatic mammal with reasonable rapidity
- . Claim tailless river creature returned at a quick pace
- Short charge by furry animal turning quite quickly
- Way to play with voice, capturing large bird
- Austin welcomes sextet, regularly presenting musical piece
- Music instruction to let Elgar go with a swing
Buddy explains!

The straight part
We know that "quite briskly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case quite briskly = allegretto.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 ALLEGRETTO mean?
ALLEGRETTO noun- A movement in this time.
- To be played rather fast and lively.