Cheer near the construction (7)
I believe the answer is:
hearten
'cheer' is the definition.
('hearten' can be a synonym of 'cheer')
'near the construction' is the wordplay.
'construction' is an anagram indicator (answer constructed from certain letters).
'near'+'the'='nearthe'
'nearthe' anagrammed gives 'HEARTEN'.
(Other definitions for hearten that I've seen before include "Inspirit" , "Encourage or cheer in cardiac manner" , "Buoy" , "Make cheerful" , "Inspire anew" .)