Trials just finished on new sets (5)
I believe the answer is:
tests
'trials' is the definition.
('test' can be a synonym of 'trial')
'just finished on new sets' is the wordplay.
'just finished' becomes 't' (I can't justify this - if you can you should give a lot more credence to this answer).
'on' indicates putting letters inside.
'new' indicates anagramming the letters.
'sets' anagrammed gives 'ests'.
't' going within 'ests' is 'TESTS'.
Can you help me to learn more?
(Other definitions for tests that I've seen before include "Trials for cricketers" , "International cricket matches" , "Puts to proof at cricket matches" , "Examinations, trials" , "Tries out" .)