Bishop settled on that man being unconcerned (6)
I believe the answer is:
blithe
'unconcerned' is the definition.
('blithe' can be similar in meaning to 'unconcerned')
'bishop settled on that man' is the wordplay.
'bishop' becomes 'b' (abbreviation used in chess).
'settled' becomes 'lit' (I've seen this before).
'on' means one lot of letters go next to another.
'that man' becomes 'he'.
'b'+'lit'+'he'='BLITHE'
'being' is the link.
(Other definitions for blithe that I've seen before include "Carefree and casual" , "Happy, without worries or cares" , "Careless" , "Happy and maybe casual" , "Gay" .)