Dispense with the commercial political leader (10)

I believe the answer is:

administer

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'dispense' is the definition.
(synonyms)

'with the commercial political leader' is the wordplay.
'with' means one lot of letters go next to another.
'the commercial' becomes 'ad' (abbreviation for advertisement).
'political leader' becomes 'minister' (minister is a kind of leader. I am not sure about the 'political' bit.).
'ad' put next to 'minister' is 'ADMINISTER'.

Can you help me to learn more?

(Other definitions for administer that I've seen before include "Oversee" , "give" , "Govern" , "Organise" , "Manage" .)

I've seen this clue in the Sydney Morning Herald.

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