An animal and one dicky bird! (6)
I believe the answer is:
Here is my best explanation:
'bird' is the definition.
(pigeon is a kind of bird)
'an animal and one dicky' is the wordplay.
'an animal' becomes 'pig' (I've seen this in other clues**).
'and' means one lot of letters go next to another.
'dicky' is an anagram indicator (dicky can mean ill or non-functioning).
'one' anagrammed gives 'eon'.
(Other definitions for pigeon that I've seen before include "gull", "run, perhaps", "Common city bird", "Common town bird", "Bird (with a hole?)", "Racing bird of town and wood", "Bird of the dove family", "a visitor to Trafalgar Square", "Homer, possibly" and "Bird of towns and wood".)