Piece about Republican producing tweet (5)
I believe the answer is:
chirp
'tweet' is the definition.
(I know that tweet can be written as chirp)
'piece about republican' is the wordplay.
'piece' becomes 'chip' (chip is a kind of piece).
'about' means one lot of letters goes inside another.
'republican' becomes 'r' (abbreviation).
'chip' going around 'r' is 'CHIRP'.
'producing' is the link.
(Other definitions for chirp that I've seen before include "Tweet" , "Twitter" , "Short sharp tone of grasshopper" , "Make sound like grasshopper" , "Fledgling's sound" .)