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Can a rocket run only on hydrogen and oxygen? Could a rocket, rover, and electrolysis machine be sent to either of Mars' poles to excavate ice to make hydrogen and oxygen to refuel an unmanned rocket?

What would be the benefit of not hauling the return fuel?
Could the travel time be shortened if fuel was waiting on Mars?

Can a machine process and run on $CO_2$ and $H_2O$ for an $O_2$ by product?

https://chemistry.stackexchange.com/questions/112136/what-is-the-ideal-temperature-to-crack-water

It absolutely could! First of all, water can be split in to hydrogen and oxygen, which can be enough to launch a rocket. Hydrogen requires a very low temperature, and the rocket engine doesn't have as much thrust as other options out there, but it is the same fuel that ran the Space Shuttle main engine, among others.

Water and carbon dioxide, easily available in the atmosphere of Mars, can be used to make Methane and Oxygen, which is a fantastic rocket fuel. This is the central premise of the book "The Case for Mars" by Robert Zubrin, and features critically in SpaceX's plans to colonize Mars. This is easier to store, and gives a higher thrust compared to hydrogen/ oxygen. It does take a lot of power to do so, however, and that might be an issue.

The main benefit is it requires a whole lot less cargo to be carried to Mars. If you can refuel a rocket on Mars, you can make a mission there reasonably priced, compared to the HUGE price that it would take to do one without in-situ resource utilization.

Have a look at the wikipedia article for "In situ resource utilization". This is exactly what you're talking about, creating fuel on another planetary body.

https://en.wikipedia.org/wiki/In_situ_resource_utilization

Note that this is a fundamental part of "Mars Direct", one of the most popular ideas for a manned mission to Mars.

https://en.wikipedia.org/wiki/Mars_Direct

The way this would work is actually you would take liquid hydrogen with you, and combine this with carbon-dioxide from the atmosphere to make Methane (CH4) and Oxygen (O2). This is known as the Sabatier Reaction.

https://en.wikipedia.org/wiki/Sabatier_reaction

This can be done, as Robert Zubrin demonstrated in 1993, building an example system that ran at 94% efficiency for $47,000. The advantage of this is that Hydrogen makes up only 5% by weight of methane, so using 6 tonnes of hydrogen you can create more than 100 tonnes of fuel in the end.

This makes the mission much easier to handle, as the more fuel you need to carry with you to get there, the heavier your craft needs to be.

There are also engines known as a "Microwave Electro-Thermal (MET) thrusters". (https://www.researchgate.net/publication/3166116_The_microwave_electro-thermal_MET_thruster_using_water_vapor_propellant). These are primarily aimed for in vacuum engines, but as others have pointed out, Hydrolox fuel has been used throughout history for in-atmosphere engines too.