Rider in pursuit, a rein loosely hanging around (10)
I believe the answer is:
equestrian
'rider' is the definition.
(equestrian is a kind of rider)
'pursuit a rein loosely hanging around' is the wordplay.
'pursuit' becomes 'quest' ('quest' can be a synonym of 'pursuit'**).
'loosely' indicates anagramming the letters.
'hanging around' is an insertion indicator.
'a'+'rein'='arein'
'arein' anagrammed gives 'erian'.
'quest' put within 'erian' is 'EQUESTRIAN'.
'in' acts as a link.
(Other definitions for equestrian that I've seen before include "Jockey" , "Horseman" , "Horse rider" , "Relating to horse riding" , "Person on horseback" .)