Investigate key vote, centre having fallen away (5)
I believe the answer is:
delve
'investigate' is the definition.
'key vote centre having fallen away' is the wordplay.
'key' becomes 'del' (del key on a keyboard).
'centre having fallen away' says to hollow out the word (remove centre letters).
'vote' with its middle taken out is 've'.
'del'+'ve'='DELVE'
(Other definitions for delve that I've seen before include "Reach down and search" , "Make painstaking enquiries (into something)" , "Dig up - search" , "Intensively research (into something)" , "probe deeply" .)