The most common and most likely answer to this crossword clue is the 9 letter word DUTIFULLY
I've seen this crossword clue in The Daily Telegraph
D
U
T
I
F
U
L
L
Y
Buddy explains!

The straight part
We know that "in a respectful way" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case in a respectful way = dutifully.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.
Crossword Answer definitions
DUTIFULLY adverb- With a regard to duty; in a dutiful manner.