Local objection means there won't be any gym (4)
I believe the answer is:
nope
'local objection means there won't' is the definition.
I can't judge whether this definition defines the answer.
'any gym' is the wordplay.
'any' becomes 'no' (I can't explain this - if you can you should believe this answer much more).
'gym' becomes 'pe' (short for Physical Education).
'no'+'pe'='NOPE'
'be' acts as a link.
Can you help me to learn more?
(Other definitions for nope that I've seen before include "You must be joking" , "not with stress" , "definitely not!" , "Not on your life!" , "Emphatically not!" .)