Their having lost the chess piece is annoying (6)
I believe the answer is:
irking
'is annoying' is the definition.
(I've seen this before)
'their having lost the chess piece' is the wordplay.
'having lost' indicates named letters should be taken away.
'chess piece' becomes 'king' (king is a kind of chess piece).
'their' with 'the' taken away is 'ir'.
'ir'+'king'='IRKING'
(Other definitions for irking that I've seen before include "Annoying, irritating" , "Irritating, vexing" , "Disgusting" , "Annoying, bothering" .)