The most common and most likely answer to this crossword clue is the 9 letter word DOMINATES
I've seen this crossword clue in The Daily Telegraph
D
O
M
I
N
A
T
E
S
More clues leading to the result DOMINATES
Buddy explains!

The straight part
We know that "Projects one’s personality over" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case projects one’s personality over = dominates.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 DOMINATES mean?
DOMINATES verb- To govern, rule or control by superior authority or power
- To exert an overwhelming guiding influence over something or someone
- To enjoy a commanding position in some field
- To overlook from a height.