Music centre: last of records on it (5)
I believe the answer is:
score
'music' is the definition.
(score is a kind of music)
'centre last of records on it' is the wordplay.
'centre' becomes 'core' (core is a kind of centre).
'last of' indicates one should take the final letters.
'on it' says to put letters next to each other.
The last letter of 'records' is 's'.
'core' put after 's' is 'SCORE'.
(Other definitions for score that I've seen before include "Result" , "Gain a point (20 of them?)" , "Musical composition in written form" , "Music from film" , "Mark; twenty" .)