The most common and most likely answer to this crossword clue is the 4 plus 4 letter word HAND OVER
I've seen this crossword clue in The Guardian
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More clues leading to the result HAND OVER
Buddy explains!

The straight part
We know that "Relinquish" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case relinquish = hand over.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 HAND OVER mean?
HAND OVER noun- The transference of authority, control, power or knowledge from one agency to another, or from one state to another.
- The information passed on in such a case.
- (cellular telecommunications) the process of transferring an ongoing call or data session from one channel connected to the core network to another channel.
- (satellite telecommunications) the process of transferring satellite control responsibility from one earth station to another without loss or interruption of service.
- The transfer of goods from the dealer to the purchaser, often of illegal goods.