The most common and most likely answer to this crossword clue is the 3 plus 3 plus 6 letter word GET THE NEEDLE
G
E
T
T
H
E
N
E
E
D
L
E
Buddy explains!
The straight part
We know that "feel annoyed" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case feel annoyed = get the needle.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.