Part of record - one's poem (7)
I believe the answer is:
episode
'part' is the definition.
(I have seen 'Part of serial' mean 'episode' so perhaps 'part' could also mean 'episode')
'record one's poem' is the wordplay.
'record' becomes 'EP' (abbreviation for extended play record).
'one's' becomes 'i's' (Roman numeral).
'poem' becomes 'ode' (ode is a type of poem).
'ep'+'is'+'ode'='EPISODE'
'of' is the link.
(Other definitions for episode that I've seen before include "Story instalment" , "One programme of a series" , "Programme of a serial" , "A chapter or incident" , "from longer work" .)