The most common and most likely answer to this crossword clue is the 5 plus 2 letter word STUFF IT
I've seen this crossword clue in Financial Times
S
T
U
F
F
I
T
More clues leading to the result STUFF IT
Buddy explains!
The straight part
We know that "What to do with turkey?/I give up!" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case what to do with turkey?/i give up! = stuff it.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.