The most common and most likely answer to this crossword clue is the 3 plus 3 plus 6 letter word ARE YOU DECENT
I've seen this crossword clue in The Guardian
A
R
E
Y
O
U
D
E
C
E
N
T
Buddy explains!
The straight part
We know that "cover-up inquiry" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case cover-up inquiry = are you decent.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.