The most common and most likely answer to this crossword clue is the 7 letter word FANTAIL
I've seen this crossword clue in The Daily Telegraph
F
A
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T
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More clues leading to the result FANTAIL
Buddy explains!

The straight part
We know that "Pigeon" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case pigeon = fantail.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 FANTAIL mean?
FANTAIL noun- Any of several birds, of the genus Rhipidura, from Asia, Australia and New Zealand.
- Any of several domestic varieties of pigeon having a fan-shaped tail.
- Any of several goldfish having a large fan-shaped tail.
- An overhanging deck at the stern of a ship
- A small windmill mounted at right angles to the sails, at the rear of the windmill, used to turn the cap automatically to bring it into the wind.
- A brimmed hat with just the back of the brim turned up.