He has his favourite period piece (5)
I believe the answer is:
peter
'piece' is the definition.
(peter is a kind of piece)
'he has his favourite period' is the wordplay.
I cannot quite see how this works, but
'he' could be 'pete' and 'pete' is found in the answer.
A single letter 'r' remains which might be clued in a way I don't understand.
This explanation may well be incorrect...
Can you help me to learn more?
(Other definitions for peter that I've seen before include "- Grimes (Britten)" , "Safe; saint" , "and 14 dn Latest actor to play Doctor Who" , "Boy's name (maybe blue)" , "Lord -- Wimsey (Sayers)" .)