The most common and most likely answer to this crossword clue is the 4 letter word COIF
C
O
I
F
Buddy explains!
The straight part
We know that "nun's cap" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case nun's cap = coif.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 COIF mean?
COIF noun- A hairdo.
- A hood; a close-fitting cap covering much of the head, widespread until the 18th century; after that worn only by small children and country women.
- An item of chain mail headgear.
- An official headdress, such as that worn by certain judges in England.
- To style or arrange hair.