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I know that car engines rely on expanding gas to cause pressure to force the piston down. My question is during the compression state, why can't they add more gas/air to cause more pressure and force the piston down?
Why do they have to combust it?
Can't the same result happen from just adding more particles?
Why combustion?
What you are proposing has been done a few times: there are cars that run on compressed air.
The problem with that is that compressed air has a low energy density, comparable to lead-acid batteries and 30 times lower than petrol or diesel. So compressed air cars have short range. They're also harder to refuel than an electric car. This combination means there are no reasons to choose compressed air over electric power or internal combustion.
The amount of energy required to force that amount of gas into the cylinder is immense. Causing a large enough pressure difference to force the piston down and drive the engine, and also the vehicle, would take a huge amount of energy. So they question isn't so much why can't they, but where would that energy come from?
The energy in a standard internal combustion engine comes from the fuel, whether thats ethanol, gasoline, or diesel. The fuel combusts and releases a large amount of energy in the form of heat, which causes the air to heat and expand. The heated air in the same volume of space creates a high pressure, which forces the piston down.
What's advantageous about internal combustion engines is that the fuel they use has a very high energy density. That is, they have a lot of energy for the amount of space and weight they take to store. Electric motors are great in the sense that they are incredibly efficient. While the batteries are currently are no where close to the energy density of more traditional fuels, there is much less energy lost in their use to heat and friction.
In your proposed system, the gas injector would be doing all the work of creating the pressure. It would need to be seriously strong, and have a large amount of energy behind it to do all this work. It's not impossible, but it would likely be much less efficient than anything that currently exists.
They do; it's called forced induction and is commonplace across most modern vehicles. The use of a compressor, typically a Turbocharger (although sometimes a Supercharger) means air at higher than atmospheric pressure is forced into the inlet manifold alongside additional fuel to keep the mixture at circa 14:1. Higher pressure turbo systems pair the compressor up with an intercooler (like a big radiator) to cool the charge because cold air has a higher density (thus a higher oxygen count) so you get more "bang for your buck" as they say.
