Fool ran them ragged, being outside persecution (10)
I believe the answer is:
harassment
'persecution' is the definition.
(I've seen this before)
'fool ran them ragged being outside' is the wordplay.
'fool' becomes 'ass' (both can mean a stupid person).
'ragged' is an anagram indicator.
'being outside' is an insertion indicator (some letters go outside others).
'ran'+'them'='ranthem'
'ranthem' is an anagram of 'harment'.
'ass' inserted into 'harment' is 'HARASSMENT'.
(Other definitions for harassment that I've seen before include "Intimidating behaviour" , "Persecution" , "Means trash (anag.)" , "Constant intimidation" , "Perhaps stalking" .)