Rider's hesitation about expedition in surrounding area (10)
I believe the answer is:
equestrian
'rider's' is the definition.
(equestrian is a kind of rider)
'hesitation about expedition in surrounding area' is the wordplay.
'hesitation' becomes 'er' (sound made while hesitating - um, er).
'about' is an insertion indicator.
'expedition' becomes 'quest' (I've seen this in other clues).
'surrounding' is an insertion indicator.
'area' becomes 'a' (maths abbreviation).
'er' placed around 'quest' is 'equestr'.
'in' enclosing 'a' is 'ian'.
'equestr'+'ian'='EQUESTRIAN'
(Other definitions for equestrian that I've seen before include "Relating to horse riding" , "Jockey" , "Horse rider" , "Horse-rider" , "Horseman" .)