A burger, but hold the filling - Bluebird's full (5)
I believe the answer is:
abrim
'a burger' is the definition.
I don't know anything about this answer so I can't judge whether this works.
'hold the filling bluebird's full' is the wordplay.
'hold' becomes 'gap'.
'the filling' indicates the central letters (the letters which fill the word).
'bluebird's full' becomes 'brim' (I can't explain this - if you can you should give a lot more credence to this answer).
The middle of 'gap' is 'a'.
'a'+'brim'='ABRIM'
'but' acts as a link.
Can you help me to learn more?
(Other definitions for abrim that I've seen before include "Completely full" , "Spilling over" .)