The most common and most likely answer to this crossword clue is the 4 plus 5 letter word LOSE HEART
I've seen this crossword clue in The Daily Telegraph
L
O
S
E
H
E
A
R
T
More clues leading to the result LOSE HEART
- Fail to win compassion, become disillusioned
- Fall in love and start to despair?
- Try to enter small room with opening barred and give up
- Jockey let a horse give up
- Give up the ales or rum?
- Become despondent as Bethany would if husband left
- Gather in secret without initial despair
- See group surrounding judge – become disillusioned
- Fall in love, yet become discouraged
- Become despondent as the role gets changed
Buddy explains!

The straight part
We know that "become despondent" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case become despondent = lose heart.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.