Tart having a lot of fun with doctor and government minister (7)
I believe the answer is:
profumo
'doctor and government minister' is the definition.
I know nothing about this answer so I cannot tell whether it can be defined by this definition.
'tart having a lot of fun' is the wordplay.
'tart' becomes 'promo' (I can't justify this - if you can you should give a lot more credence to this answer).
'having' means one lot of letters goes inside another.
'a lot of' means to remove the last letter (most of the word but not all of it).
'fun' with its last letter taken away is 'fu'.
'promo' enclosing 'fu' is 'PROFUMO'.
'with' acts as a link.
Can you help me to learn more?
(Other definitions for profumo that I've seen before include "Politician in 1960s scandal" , "penitent politician" , "MP" , "John --, minister who resigned in 1963" .)