Hybrid motors using solid fuels (plastic, rubber, cardboard, even sausage) and liquid oxidizer (typically nitrous oxide) are pretty common in hobby rocketry. These can be trivially made throttleable within a range by metering the NO2 flow. With the proper fuel core, they can be kept "smoldering" for shutdown / relight by maintaining a trickle of oxidizer to keep the fuel surface at combustion temperature.