The most common and most likely answer to this crossword clue is the 9 letter word TEASINGLY
I've seen this crossword clue in Financial Times
T
E
A
S
I
N
G
L
Y
Buddy explains!
The straight part
We know that "in irritating manner" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case in irritating manner = teasingly.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.