Bird and horse knackered by rodent (8)
I believe the answer is:
titmouse
'bird' is the definition.
(titmouse is a kind of bird)
'by rodent' is the wordplay.
'by' becomes 'tit' (I can't explain this - if you can you should believe this answer much more).
'rodent' becomes 'mouse' (mouse is a kind of rodent).
'tit'+'mouse'='TITMOUSE'
'and horse knackered' acts as a link.
This may not be correct. Some or all of it may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for titmouse that I've seen before include "Small bird" , "Small songbird" .)