Conceited weaver revolutionary artist spots astride horse (7)
I believe the answer is:
arachne
'conceited weaver' is the definition.
(mythical character)
'revolutionary artist spots astride horse' is the wordplay.
'revolutionary' says the letters should be written backwards.
'artist' becomes 'ra' (Royal Academician, member of Royal Academy of Arts).
'spots' becomes 'acne' (I've seen this before**).
'astride' means one lot of letters goes inside another.
'horse' becomes 'h' ('H' can be a synonym of 'horse').
'ra' back-to-front is 'ar'.
'acne' enclosing 'h' is 'achne'.
'ar'+'achne'='ARACHNE'
(Other definitions for arachne that I've seen before include "Greek girl who unwisely challenged Athena and ended up as a spider" , "She was transformed to a spider in Greek myth" .)