Recess seeing American kids finally getting into exercise (4)
I believe the answer is:
apse
'recess' is the definition.
(apse is a kind of recess)
'american kids finally getting into exercise' is the wordplay.
'american' becomes 'a' (common abbreviation - e.g. in organisation names).
'finally' indicates one should take the final letters.
'getting into' indicates putting letters inside.
'exercise' becomes 'pe' (Physical Education).
The last letter of 'kids' is 's'.
's' put within 'pe' is 'pse'.
'a'+'pse'='APSE'
'seeing' acts as a link.
(Other definitions for apse that I've seen before include "Church's part" , "Recess in church wall" , "Offshoot from the choir, nave or aisle in a church" , "Semicircular end of a church, behind the altar" , "Peas (anag) -- part of a church" .)