An injury, see, to display with sensitivity (7)
I believe the answer is:
acutely
'with sensitivity' is the definition.
(as in being acutely aware of something)
'an injury see' is the wordplay.
'an' becomes 'a'.
'injury' becomes 'cut' (cut is a kind of injury).
'see' becomes 'ely' (see is a place with a cathedral - e.g. Ely).
'a'+'cut'+'ely'='ACUTELY'
'to display' is the link.
(Other definitions for acutely that I've seen before include "with severity" , "Critically" , "Extremely" , "Pointedly" , "Sharply" .)