Any hydrocarbon engine will have hydrogen in the thrust chamber, and in the gas generator. They all run a bit fuel rich to maximize Isp, and that will mean the hot partially burned gas has some molecular hydrogen in it.
What makes hydrogen engines more expensive is the low density of LH2, which greatly increases the pumping power needed to bring the propellant up to pressure for injection into the thrust chamber.
What makes hydrogen engines more expensive is the low density of LH2, which greatly increases the pumping power needed to bring the propellant up to pressure for injection into the thrust chamber.