Part of bridle gripped by the mouth's going round (7)
I believe the answer is:
orbital
'going round' is the definition.
(I've seen this before)
'part of bridle gripped by the mouth's' is the wordplay.
'part of bridle' becomes 'bit' ('bit' is part of 'bridle').
'gripped by' is an insertion indicator.
'the mouth's' becomes 'oral' (relating or belonging to the mouth).
'bit' inserted within 'oral' is 'ORBITAL'.
(Other definitions for orbital that I've seen before include "Like a ring road" , "Like a sort of road" , "Going round the world" , "Boil tar (anag)" , "Following a path around a heavenly body" .)