Bishop unconscious in period of illness (4)
I believe the answer is:
bout
'period of illness' is the definition.
(as in a bout of flu)
'bishop unconscious' is the wordplay.
'bishop' becomes 'B' (abbreviation for bishop in chess).
'unconscious' becomes 'out' (similar in meaning).
'b'+'out'='BOUT'
'in' is the link.
(Other definitions for bout that I've seen before include "Wrestling/boxing match" , "Short period of intense activity" , "(Drinking) session" , "Spell of eg drinking" , "Wrestling contest" .)