Agnes loses heart and grows old (4)
I believe the answer is:
ages
'grows old' is the definition.
('age' can be a synonym of 'grow old')
'agnes loses heart' is the wordplay.
'loses heart' suggests removing the centre.
'agnes' with its middle taken out is 'AGES'.
'and' acts as a link.
(Other definitions for ages that I've seen before include "Yonks" , "Awfully long time" , "... of Man, according to WB Yeats" , "Sage (anag)" , "A long time (colloq.)" .)