The most common and most likely answer to this crossword clue is the 6 plus 3 letter word TUCKER OUT
T
U
C
K
E
R
O
U
T
Buddy explains!
The straight part
We know that "Exhaust" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case exhaust = tucker 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.