To put under estimated second concerns (9)
I believe the answer is:
interests
'concerns' is the definition.
(I know this)
'to put under estimated second' is the wordplay.
'to put under' becomes 'inter' (I've seen this in another clue).
'estimated' becomes 'est' (abbreviation).
'second' becomes 's' (common abbreviation).
'inter'+'est'+'s'='INTERESTS'
(Other definitions for interests that I've seen before include "Financial involvements or matters of curiosity" , "Involvements, pastimes" , "Hobbies or preoccupations" , "Advantages" , "Pursuits of triteness" .)