Governess Grey having lost heart matures (4)
I believe the answer is:
ages
'matures' is the definition.
('age' can be a synonym of 'mature')
'governess grey having lost heart' is the wordplay.
'governess grey' becomes 'age' (I am not sure about this - if you are sure you should believe this answer much more).
'having' means one lot of letters go next to another.
'lost' becomes 'atsea' (I've seen this before).
'heart' means to look at the middle letters.
The middle letter of 'atsea' is 's'.
'age'+'s'='AGES'
Can you help me to learn more?
(Other definitions for ages that I've seen before include "(What seems) a very long time" , "Grows old like sage" , "... of Man, according to WB Yeats" , "Epochs" , "Yonks" .)