Teach to run a posh estate
I believe the answer is:
chateau
'estate' is the definition.
'teach to run a posh' is the wordplay.
'to run' is an anagram indicator.
'posh' becomes 'u' (abbreviation for upper-class - opposite to non-u).
'teach' with letters rearranged gives 'chate'.
'chate'+'a'+'u'='CHATEAU'
(Other definitions for chateau that I've seen before include "A grand French Mansion" , "Country residence" , "French manor house" , "French castle or manor" , "Impressive dwelling in France" .)