There are two (broad) approaches that a car manufacturer can take for a “hybrid” gas electric vehicle.
One is where the gas engine and electric motor can both drive the wheels of the car. So, in a way, independently as you put it. The electric motor and battery in this design is usually small so can probably only power the car without the gas engine at slow speeds. The electric motor is designed to save gas (say cruising) or for additional power during overtaking – which allows the manufacturer to put in a smaller gas engine. Think of it as a gas powered car with electric motor supplementing.
The second approach is one where the car can only be driven by the electric motor. The gas engine is used to generate electricity to charge the batteries. In this case, to your question, the car cannot be run on the gas engine. The electric motors have to be more powerful and typically these kind of hybrids have larger battery packs too. This is an electric car with a gas engine range extender.
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