The most common and most likely answer to this crossword clue is the 7 letter word JARRING
I've seen this crossword clue in The Guardian
J
A
R
R
I
N
G
Buddy explains!

The straight part
We know that "annoying" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case annoying = jarring.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 JARRING mean?
JARRING verb- To preserve (food) in a jar.
- To knock, shake, or strike sharply, especially causing a quivering or vibrating movement.
- To harm or injure by such action.
- To shock or surprise.
- To act in disagreement or opposition, to clash, to be at odds with; to interfere; to dispute, to quarrel.
- To (cause something to) give forth a rudely tremulous or quivering sound; to (cause something to) sound discordantly or harshly.
- To quiver or vibrate due to being shaken or struck.
- Of the appearance, form, style, etc., of people and things: to look strangely different; to stand out awkwardly from its surroundings; to be incongruent.
- Storage in jars.
- A feeling or movement that jars or jolts.
- That jars (clashes or disagrees); incongruous, conflictful.