Horse runs about after snake (5)
I believe the answer is:
cobra
'snake' is the definition.
(I've seen this before)
'horse runs about after' is the wordplay.
'horse' becomes 'cob' (cob is a kind of horse).
'runs' becomes 'r' (cricket abbreviation).
'about' becomes 'a' (abbreviation in Chambers**).
'after' is a charade indicator (letters next to each other).
'cob'+'r'+'a'='COBRA'
(Other definitions for cobra that I've seen before include "One with hood" , "one leading charmed life?" , "Venomous hooded snake" , "dangerous striker" , "Venomous snake, may be king" .)