Man in garden determined to eradicate insect (4)
I believe the answer is:
adam
'man in garden' is the definition.
(in the Garden of Eden)
'determined to eradicate insect' is the wordplay.
'determined' becomes 'adamant' (I've seen this before).
'to eradicate' is a deletion indicator.
'insect' becomes 'ant' (ant is a kind of insect).
'adamant' with 'ant' removed is 'ADAM'.
(Other definitions for adam that I've seen before include "Faith, for example" , "expelled gardener" , "First person" , "Fellow" , "who transformed the Palladian style?" .)