Government department having little right to visit filthy home (8)
I believe the answer is:
ministry
'government department' is the definition.
(ministry is a kind of government department)
'little right to visit filthy home' is the wordplay.
'little' becomes 'mini'.
'right' becomes 'r' (common abbreviation).
'to visit' indicates putting letters inside.
'filthy home' becomes 'sty'.
'r' placed within 'sty' is 'stry'.
'mini'+'stry'='MINISTRY'
'having' is the link.
(Other definitions for ministry that I've seen before include "Government building" , "a cleric has entered" , "Priesthood" , "Government department" , "Government employees" .)