Part of city starts to slag off Home Office (4)
I believe the answer is:
soho
'part of city' is the definition.
(Soho is an area of London)
'starts to slag off home office' is the wordplay.
'starts to' suggests taking the first letters.
'home office' becomes 'HO' (abbreviation).
The initial letters of 'slag off' is 'so'.
'so'+'ho'='SOHO'
(Other definitions for soho that I've seen before include "Parts of London and New York City with the same name" , "Colourful London show-biz district" , "Colourful district in London's West End" , "W London area" , "London district known for its nightlife" .)