The most common and most likely answer to this crossword clue is the 8 letter word HARRYING
I've seen this crossword clue in Financial Times
H
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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 = harrying.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 HARRYING mean?
HARRYING verb- To plunder, pillage, assault.
- To make repeated attacks on an enemy.
- To strip, lay waste, ravage.
- To harass, bother or distress with demands, threats, or criticism.
- The act of one who harries or harasses.