The most common and most likely answer to this crossword clue is the 6 letter word CHADOR
I've seen this crossword clue in The Guardian
C
H
A
D
O
R
Buddy explains!

The straight part
We know that "A cover-up" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case a cover-up = chador.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 CHADOR mean?
CHADOR noun- A loose robe, made from a single cloth, worn as a combination head covering, veil and shawl by Muslim women, especially in Iran.