The most common and most likely answer to this crossword clue is the 9 letter word PRIVATION
I've seen this crossword clue in The Guardian
P
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More clues leading to the result PRIVATION
Buddy explains!

The straight part
We know that "Austerity" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case austerity = privation.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 PRIVATION mean?
PRIVATION noun- The state of being deprived of or lacking an attribute formerly or properly possessed; the loss or absence of such an attribute.
- The state of being very poor, and lacking the basic necessities of life.
- The act of depriving someone of such basic necessities; deprivation.
- Degradation or suspension from an office.