The most common and most likely answer to this crossword clue is the 5 letter word HOMES
I've seen this crossword clue in Financial Times
H
O
M
E
S
Buddy explains!
The straight part
We know that "what pigeon does" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case what pigeon does = homes.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 HOMES mean?
HOMES noun- An informal term of address, like man or dude.
- A dwelling.
- One’s native land; the place or country in which one dwells; the place where one’s ancestors dwell or dwelt.
- The locality where a thing is usually found, or was first found, or where it is naturally abundant; habitat; seat.
- A focus point.
- (of animals) To return to its owner.
- (always with "in on") To seek or aim for something.
- A directory that contains a user's files.