Bird is almost out of breath (6)
I believe the answer is:
puffin
'bird' is the definition.
(puffin is a kind of bird)
'almost out of breath' is the wordplay.
'almost' means to remove the last letter (most of the word but not all of it).
'out of breath' becomes 'puffing' (I've seen this in another clue).
'puffing' with its final letter removed is 'PUFFIN'.
'is' is the link.
(Other definitions for puffin that I've seen before include "Seabird with brightly coloured bill - sounds out of breath" , "Bright-beaked sea bird" , "Seabird (breathin' heavily?)" , "Bright-billed bird" , "Seabird with parrot-like beak" .)