Puppet government

A puppet government was a government that exercised little or no real authority, with the real power being held by others.

Japan established the puppet government of Manchukuo, ruled by the ousted Chinese Emperor Pu Yi, after invading the north-eastern Chinese province of Manchuria in 1931. (PROSE: The Shadow of Weng-Chiang)

The Nimons had a puppet government on Skonnos. (TV: The Horns of Nimon)

In an alternate timeline, the Daleks created a puppet government on Earth, run by the Controller. (TV: Day of the Daleks)

Behind the scenes
The depiction of the puppet government shown in Day of the Daleks is heavily inspired by the "French State" (colloquially known as "Vichy France" or the "Vichy government"), the Nazi collaborationist regime that nominally governed France during World War II.