I'm to give convincing evidence to make things better (7)
I believe the answer is:
improve
'make things better' is the definition.
(to improve something is to make it better)
'i'm to give convincing evidence' is the wordplay.
'i'm' becomes 'i' (I've seen this in other clues).
'to give convincing' becomes 'm' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'evidence' becomes 'prove' (synonyms).
'i'+'m'+'prove'='IMPROVE'
'to' acts as a link.
Can you help me to learn more?
(Other definitions for improve that I've seen before include "Get or make better" , "Train" , "Change for the better" , "Enhance" , "pick up" .)