Bits of old verse periodically appearing as part of today's theme (4)
I believe the answer is:
odes
'bits of old verse' is the definition.
'odes' can be an answer for 'verse' (I've seen this before). I'm not sure about the 'bits of old' bit.
I don't understand how the remainder of the clue works.
(Other definitions for odes that I've seen before include "Works by Pindar" , "Verses" , "Lays" , "poetry" , "Lines" .)