The bird, within half a minute, catches the fleeing rat (6)
I believe the answer is:
martin
'the bird' is the definition.
(martin is a kind of bird)
'half a minute catches the fleeing rat' is the wordplay.
'half a' suggests halving the letters.
'catches' indicates putting letters inside.
'the fleeing rat' becomes 'art' (I can't explain this - if you can you should believe this answer much more).
'minute' with half the letters taken is 'min'.
'min' enclosing 'art' is 'MARTIN'.
'within' is the link.
I am not very happy about this link. It may be part of another bit of the clue.
Can you help me to learn more?
(Other definitions for martin that I've seen before include "Male name, bird" , "Tweeter" , "Bird or boy's name" , "Bird of the swallow family" , "25 across [CRANE]" .)