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)
After the defeat of France during World War II, the nation and its African colonies was run by the Vichy government, which was subject to Nazi Germany. (PROSE: The Turing Test)
The Nimons had a puppet government on Skonnos. (TV: The Horns of Nimon)
The Fourth Doctor claimed that Professor Kellman would become a puppet dictator of the solar system under the Cybermen once they invaded. However, the Cyber-Leader had no intention of fulfilling this term of their alliance, while Kellman was working against them as a Vogan double agent anyway. (TV: Revenge of the Cybermen)
The government of Winter was essentially a puppet of the Dalek Empire, with the Grand Marshall of Winter willingly deceiving their own people into serving the Daleks. (AUDIO: The Dalek Occupation of Winter)
Alternate timelines
In an alternate timeline in which Nazi Germany won World War II, Occupied Britain became a puppet state run by a National Socialist Parliament under Oswald Mosley, and King Edward VIII returned to the throne. The real UK government set itself up in exile in Canada. (PROSE: Timewyrm: Exodus)
In another alternate timeline, the Daleks created a puppet government on Earth, run by the Controller. (TV: Day of the Daleks)
Parallel universes
In Germania I, the United States government became a Nazi puppet, with US-SS members working in the Senate.
The Greater German Reich also effectively set up puppet governments in other Germanias deemed to have strayed from pure Nazi ideals, such as Germania LXVIII. (PROSE: Warlords of Utopia)