The most common and most likely answer to this crossword clue is the 6 letter word WAIVED
I've seen this crossword clue in The Sunday Telegraph
W
A
I
V
E
D
Buddy explains!

The straight part
We know that "Gave up" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case gave up = waived.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 WAIVED mean?
WAIVED verb- To relinquish (a right etc.); to give up claim to; to forego.
- To put aside, avoid.
- To outlaw (someone).
- To abandon, give up (someone or something).
- To move from side to side; to sway.
- To stray, wander.