The most common and most likely answer to this crossword clue is the 9 letter word IMPUDENCE
I've seen this crossword clue in The Guardian
I
M
P
U
D
E
N
C
E
More clues leading to the result IMPUDENCE
- Gall bladder finally removed in folly
- Cheek
- Cheek reddens primarily from lack of discretion
- Thoughtlessness right away leads to rudeness
- Audacity shows lack of judgement – not right
- Dessert in pieces in cool sauce
- Lack of caution cutting right cheek
- Flippancy is thoughtlessness without a trace of respect
- My complexion’s purplish around half-hidden cheek
- Carelessness evident, having cut right cheek
Buddy explains!

The straight part
We know that "rudeness" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case rudeness = impudence.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 IMPUDENCE mean?
IMPUDENCE noun- The quality of being impudent, not showing due respect.
- Impudent language, conduct or behavior.