The most common and most likely answer to this crossword clue is the 4 plus 6 letter word STAR STRUCK
I've seen this crossword clue in The Daily Telegraph
S
T
A
R
S
T
R
U
C
K
Buddy explains!
The straight part
We know that "deeply impressed" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case deeply impressed = star struck.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 STAR STRUCK mean?
STAR STRUCK adjective- Under the malevolent influence of the stars or of one's horoscope
- Fascinated and/or obsessed by fame or by celebrities; awed by an encounter with a celebrity