The most common and most likely answer to this crossword clue is the 9 letter word CATHERINE
C
A
T
H
E
R
I
N
E
More clues leading to the result CATHERINE
- Girl jazz fan at home, in her element initially
- Jazz fan, the fellow crossing Ireland
- Circular window from Tate? London Eye?
- Cruel act by female protagonist, heartless queen
- Jazz fan, the chap going round Ireland to see girl
- She may take the wheel with dazzling effect
- She's running in the race
- Violent act by a heartless protagonist like miss Deneuve, perhaps
- Girl with unusually nice heart
- Critic, at her ineptest, collars Duchess
Buddy explains!

The straight part
We know that "miss Deneuve, perhaps" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case miss deneuve, perhaps = catherine.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.