Rider's request to cycle in surrounding area (10)
I believe the answer is:
equestrian
'rider's' is the definition.
(equestrian is a kind of rider)
'request to cycle in surrounding area' is the wordplay.
'to cycle' indicates an anagram.
'surrounding' indicates putting letters inside.
'area' becomes 'a' (maths abbreviation).
'request' anagrammed gives 'equestr'.
'equestr'+'in'='equestrin'
'equestrin' placed around 'a' is 'EQUESTRIAN'.
(Other definitions for equestrian that I've seen before include "Relating to horse riding" , "Horse rider" , "Horse-rider" , "Person on horseback" , "Horseman" .)