Postpone, to fly the old in (4,4)
I believe the answer is:
hold over
'postpone' is the definition.
(I know that postpone can be written as hold over)
'fly the old in' is the wordplay.
'fly' becomes 'hover' (hovering is a kind of flying).
'in' indicates putting letters inside.
'hover' placed around 'old' is 'HOLD OVER'.
'to' acts as a link.
(Other definitions for hold over that I've seen before include "... postpone" , "Defer action on" , "influence on another" .)