The most common and most likely answer to this crossword clue is the 5 plus 2 plus 3 letter word STICK IT OUT
I've seen this crossword clue in The Times
S
T
I
C
K
I
T
O
U
T
More clues leading to the result STICK IT OUT
Buddy explains!
The straight part
We know that "Don't give up" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case don't give up = stick it out.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.