Finally, feeling poorly, you get a breather (4)
I believe the answer is:
gill
'a breather' is the definition.
(part of a fish which breathes)
'finally feeling poorly' is the wordplay.
'finally' indicates one should take the final letters.
'poorly' becomes 'ill' (similar in meaning).
The final letter of 'feeling' is 'g'.
'g'+'ill'='GILL'
'you get' acts as a link.
(Other definitions for gill that I've seen before include "ravine" , "Breathing organ in fish" , "Fish's breathing organ" , "Six nips" , "Fish's breathing apparatus" .)