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