In the finish yours truly gets to improve (5)
I believe the answer is:
emend
'improve' is the definition.
(emending is a kind of improving)
'in the finish yours truly' is the wordplay.
'in' indicates putting letters inside.
'the finish' becomes 'end' (I've seen this in other clues).
'yours truly' becomes 'me'.
'end' placed around 'me' is 'EMEND'.
'gets to' is the link.
(Other definitions for emend that I've seen before include "Edit, improve" , "Revise (a text) (E, not A)" , "would appear better" , "Make correction in written material" , "Improve by editing" .)