The most common and most likely answer to this crossword clue is the 5 letter word WEARY
I've seen this crossword clue in The Times
W
E
A
R
Y
More clues leading to the result WEARY
- Suspicious about vehicle's second exhaust
- Tired wife inclined to cry when short of time
- Careful to keep in the final beat
- Clothes by end of stairway – ready for bed?
- Tired and cautious about Spain
- European cautious about getting tired
- Tiresome curse for the Scots, suspicious about English
- Cautious about minimal energy – when one’s this?
- Confederate’s end in battle, Yankee’s beat
- Tired clothes ending in laundry
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 = weary.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 WEARY mean?
WEARY verb- To make or to become weary.
- Having the strength exhausted by toil or exertion; tired; fatigued.
- Having one's patience, relish, or contentment exhausted; tired; sick.
- Expressive of fatigue.
- Causing weariness; tiresome.