Daughter leaves older relative badly needing a piece of fruit (10)

'fruit' is the definition.
(granadilla is a kind of fruit)

'daughter leaves older relative badly needing a piece' is the wordplay.
I cannot really see how this works, but
'daughter' could be 'd' (genealogical abbreviation) and 'd' is found within the answer.
'relative' could be 'gran' (short for grandmother) and 'gran' is found within the answer.
'badly' could be 'ill' ('ill' can be adverbial as in 'ill-advised') and 'ill' is found in the answer.
'a' is within the answer.
A single letter 'a' remains which might be clued in a way I don't understand.
This may be the basis of the clue (or it may be nonsense).

'of' is the link.

Can you help me to learn more?