Past attempt to one's heart's desire (1,4)
I believe the answer is:
a gogo
'to one's heart's desire' is the definition.
('a gogo' means galore or to your heart's content)
'past attempt' is the wordplay.
'past' becomes 'ago' (similar in meaning).
'attempt' becomes 'go' (go is a kind of attempt).
'ago'+'go'='A GOGO'
(Other definitions for a gogo that I've seen before include "In abundance (informal)" , "stacks" , "Small bell" , "as many as you like" , "in spades" .)