Auckland Council has been set up similar to the American system of governance. The President (Mayor) proposes the budget/vision for the Country(City) to the Senate/Congress(Council).
Under the old legacy Councils Council officers came to Council with their vision for the year ahead and the budget required. The Council and the Mayor could turn around and accept the Officers recommendations in full or in art or not at all following consultation with the public.
Now the Auckland Council with Mayor's vision and budget. A bit like the President of the US does except he asks the Congress and Senate for their endorsement or not of his programs for the country.
The US President can also of course be impeached and thrown out of office by the Senate/Congress. They tried with President Clinton (re sexual relationships) and Nixon stepped down over Watergate before he was impeached.
Question then why not have an impeachment process for a Mayor. Further as in the US the Mayor should only be able to stand for a maximum of two terms (perhaps all other elected members should only be allowed to stand for a maximum of three terms). That way we may encourage fresh ideas and faces into the system.
I am a member of Cogs (Community Organisation Grants Scheme) and can only stand for a maximum of two terms. And I am unpaid.