Bringing up dried fruit - grapefruit, initially (7)
I believe the answer is:
raising
'bringing up' is the definition.
(raising is a kind of bringing up)
'dried fruit grapefruit initially' is the wordplay.
'dried fruit' becomes 'raisin' (raisin is a kind of dried fruit).
'initially' suggests taking the first letters.
The initial letter of 'grapefruit' is 'g'.
'raisin'+'g'='RAISING'
(Other definitions for raising that I've seen before include "Growing" , "Promoting (in rank)" , "Lifting" , "Elevating" , "Bringing up" .)