The most common and most likely answer to this crossword clue is the 10 letter word TORMENTORS
I've seen this crossword clue in Financial Times
T
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More clues leading to the result TORMENTORS
Buddy explains!

The straight part
We know that "They annoy" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case they annoy = tormentors.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 TORMENTORS mean?
TORMENTORS noun- One who torments; a person, animal, or object that causes suffering.
- Something abstract that causes suffering.
- One of a pair of narrow curtains just behind the front curtain and teaser that mask the areas on the sides of the stage and can be adjusted to the desired width.
- An implement for reducing a stiff soil, resembling a harrow, but running upon wheels.
- A long meat-fork.