First, Adam left one bit unfinished, offering an excuse (5)
I believe the answer is:
alibi
'an excuse' is the definition.
(excuse given by a suspect in an investigation)
'first adam left one bit unfinished' is the wordplay.
'first adam' becomes 'a' (1st letter of 'adam').
'left one' becomes 'li' (left abbreviation plus 1).
'unfinished' means to remove the last letter.
'bit' with its last letter taken off is 'bi'.
'a'+'li'+'bi'='ALIBI'
'offering' is the link.
(Other definitions for alibi that I've seen before include "Legal excuse" , ""suspect's story"" , "Defence used by an accused person" , "Answer to charge - was somewhere else" , "Proof of not being at the scene of the crime" .)