Rat in middle of pigeonry? That’s an inspiration (5)
I believe the answer is:
erato
'an inspiration' is the definition.
(I've seen this before)
'rat in middle of pigeonry?' is the wordplay.
'in' is an insertion indicator.
'middle of' says to take the centre.
The middle letters of 'pigeonry' are 'eo'.
'rat' inserted within 'eo' is 'ERATO'.
'that's' is the link.
(Other definitions for erato that I've seen before include "One of the Muses" , "Clio's sister" , "Greek muse - orate (anag)" , "Muse of lyric love-poetry" , "Greek muse of lyric and love poetry" .)