"The King's Government must be carried on"

what is the meaning of the phrase "The King's Government must be carried on"?

Exactly what the words say. It was a maxim of the Duke of Wellington (after the defeat of Napoleon when he became Prime Minister). The British Government is called the King's (or Queen's) Government because theoretically the monarch is in charge and appoints ministers to govern on his/her behalf. Obviously, whatever the crisis, the government of the country has to continue.