City has Mayor's first bird flying in enclosure (9)
I believe the answer is:
cambridge
'city' is the definition.
(Cambridge is an example)
'mayor's first bird flying in enclosure' is the wordplay.
'first' indicates taking the first letters.
'flying' indicates anagramming the letters (letters fly around into new positions).
'in' means one lot of letters goes inside another.
'enclosure' becomes 'cage' (cage is a kind of enclosure).
The initial letter of 'mayors' is 'm'.
'm'+'bird'='mbird'
'mbird' anagrammed gives 'mbrid'.
'mbrid' going into 'cage' is 'CAMBRIDGE'.
'has' acts as a link.
(Other definitions for cambridge that I've seen before include "Light blue university?" , "University city" , "English university, rowing colour blue" , "English city" , "Old English university" .)