Search vigorously given time to find minor radical (7)
I believe the answer is:
rootlet
'minor radical' is the definition.
(radical can mean a root)
'search vigorously given time' is the wordplay.
'search vigorously' becomes 'rootle' (rootle can mean to root around or search).
'given' says to put letters next to each other.
'time' becomes 't' (abbreviation).
'rootle'+'t'='ROOTLET'
'to find' is the link.
(Other definitions for rootlet that I've seen before include "Minor growth under a plant" , "underground development" , "Little plant part" , "Small branch of a plant's "anchor"" , "part vital to growth" .)