Record one's poem for part of series (7)
I believe the answer is:
episode
'for part of series' is the definition.
(eg part of a TV series)
'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'
(Other definitions for episode that I've seen before include "Programme of a serial" , "Bout" , "Instalment of a TV series" , "Distinctive happening" , "One part of a series" .)