Name record on my back! (6)
I believe the answer is:
eponym
'name' is the definition.
(I know that eponym is a type of name)
'record on my back' is the wordplay.
'record' becomes 'EP' (abbreviation for extended play record).
'back' says the letters should be written backwards.
'my' written backwards gives 'ym'.
'ep'+'on'+'ym'='EPONYM'
(Other definitions for eponym that I've seen before include "Place-name derived from name of a person" , "Name of person after whom something is named" , "someone giving a name" , "eg 'Jane Eyre'" , "handle from place?" .)