The most common and most likely answer to this crossword clue is the 11 letter word SUBSERVIENT
I've seen this crossword clue in The Times
S
U
B
S
E
R
V
I
E
N
T
Buddy explains!

The straight part
We know that "fawning" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case fawning = subservient.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 SUBSERVIENT mean?
SUBSERVIENT adjective- Useful in an inferior capacity.
- Obsequiously submissive.