Blood type different? That's a nuisance (6)
I believe the answer is:
bother
'a nuisance' is the definition.
('bother' can be a synonym of 'nuisance')
'blood type different?' is the wordplay.
'blood type' becomes 'b' (B is an example).
'different?' becomes 'other' ('other' can be similar in meaning to 'different').
'b'+'other'='BOTHER'
'that's' is the link.
(Other definitions for bother that I've seen before include "Hassle - drat!" , "Badger" , "Annoy or pester" , "Fuss, ado" , "Fuss and annoyance" .)