Pushing dog away to get bone (4)
I believe the answer is:
shin
'bone' is the definition.
(shin is a kind of bone)
'pushing dog away' is the wordplay.
'dog' becomes 'pug' (pug is a kind of dog).
'away' indicates named letters should be taken away.
'pushing' with 'pug' taken out is 'SHIN'.
'to get' is the link.
(Other definitions for shin that I've seen before include "letter from Israel" , "Front end of the tibia" , "Cut of beef; part of leg" , "piece of beef" , "Body part" .)