The most common and most likely answer to this crossword clue is the 6 letter word HEARER
I've seen this crossword clue in The Guardian
H
E
A
R
E
R
Buddy explains!
The straight part
We know that "The woman’s covering part of body/— what is that part? (6)/what is that part?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case the woman’s covering part of body/— what is that part? (6)/what is that part? = hearer.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.