At home Hypatia’s keeping loose change for carrot (9)
I believe the answer is:
incentive
'carrot' is the definition.
(carrot and stick)
'at home hypatia's keeping loose change' is the wordplay.
'at home' becomes 'in' (similar in meaning).
'hypatia's' becomes 'I've' (where the crossword setter is 'Hypatia').
'keeping' is an insertion indicator.
'loose change' becomes 'cent' (I have seen 'Small change' mean 'cent' so perhaps 'change' could also mean 'cent'. I am not sure about the 'loose' bit.).
'in'+'ive'='inive'
'inive' placed around 'cent' is 'INCENTIVE'.
'for' is the link.
Can you help me to learn more?
(Other definitions for incentive that I've seen before include "encouragement" , "Something that motivates" , "Motivational reward" , "Motivating payment" , "Inducement" .)