Boxer, huge, almost makes excuse (5)
I believe the answer is:
alibi
'excuse' is the definition.
(excuse given by a suspect in an investigation)
'boxer huge almost' is the wordplay.
'boxer' becomes 'ali' (Muhammad Ali).
'huge' becomes 'big' (similar in meaning).
'almost' means to remove the last letter (most of the word but not all of it).
'big' with its final letter taken away is 'bi'.
'ali'+'bi'='ALIBI'
'makes' is the link.
(Other definitions for alibi that I've seen before include "Timely excuse" , "Proof one was elsewhere when crime was committed" , "Timely get-out" , "Excuse, defence" , "Legal proof of innocence" .)