The most common and most likely answer to this crossword clue is the 6 letter word PENURY
I've seen this crossword clue in The Guardian
P
E
N
U
R
Y
Buddy explains!

The straight part
We know that "destitution" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case destitution = penury.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 PENURY mean?
PENURY noun- Extreme want; poverty; destitution.
- A lack of something; a dearth.