Hear me out:
You "believe" one cup is equal to 100 miles, but based on what? Something you saw on YouTube? No disrespect intended, but how is that even an equality? It's like saying a half cup of gas gets a car one mile... very ambiguous. Which car? How old is it? What is the altitude/barometric pressure? What is the temperature? What kind of road are you driving on? How full are the tires? Are you going uphill or downhill? Is the road wet? How much does the car weigh? Is there wind?
But that's enough questions! I understand that these are averages. You told me you could be a little off, so I'm about to find out.
Pretend you have a car that gets 25mpg on gasoline. You want to go 100 miles. Thus, you'll need 4 gallons of gas to do it. A gallon of gasoline contains around 129,276,000 joules of energy, so the whole trip used 517,104,000 joules.
Your car's engine is about 19% efficient. Let us see how much of that gas went to actually getting you there and not out the exhaust as heat. 0.19*517,104,000 = 98,249,760 J. So, if your engine were 100% chemical energy-to-mechanical energy efficient, you would use 98,249,760/129,276,000 = .76 gallons of gasoline. This is 131.5mpg - as you can clearly see, 100% efficiency is absurd. Hang in there; there is a point to all this jabber.
Let us now examine how much energy is in a cup of water. First of all, the water has to be split into HHO. I will ignore this step because physics says it takes exactly the amount of energy to split the water as will be released when it burns, completely ruining the need to generate HHO at all. Therefore, let us pretend that we can split the water using no energy whatsoever.
One cup of water is 237mL and contains 13.2 moles of water since water is 18g/mole. Each mole of water decomposes to a half mole of oxygen and one mole of hydrogen, leaving you with 13.2 moles of hydrogen and 6.6 moles of oxygen. When one mole of hydrogen reacts with one half mole of oxygen to form water, 285.83kJ of energy is released. Therefore, 13.2 moles would produce 3,772,956 joules of energy. We now know how much energy is made when re-forming one cup of water from HHO.
Now plug this into the 100% efficient car model. We eliminated engine efficiency since, as you thoughtfully mentioned, "gas engines are not designed to run on HHO." Therefore we will pretend we have an HHO engine which is 100% efficient - the best-case scenario. It takes 98,249,760 J of energy to get the car 100 miles when engine efficiency is not taken into account. How much water is this? The cup of water had 3,772,956 J, so it would get you 100*(3,772,956/98,249,760) = 3.84 miles.
Yes. If your engine was 100% efficient AND it took no power to make your HHO, a cup of water would get you 3.84 miles in the typical 25mpg-sized car. (I use this 25mpg model since Stan's dune buggy would probably get about 25mpg if it ran on gasoline)
Even the most efficient heat engines found in power plants in the form of rankine-cycle turbines only barely approach
43% efficiency. Now we're down to 1.54 miles, assuming you've also come up with some magic HHO engine which is top-of-modern-technology efficient. Is your engine about as efficient as a petrol engine, 19%? Try 0.73 miles, or a bit short of 4000 feet.
Don't forget to account for the energy it would take to make the HHO in the first place, assuming you could break the laws of physics and make it with less energy that you'd eventually get out of it.
See my point?
Anyway, as for the "yes or no" answer:
Yes if you believe you can make HHO with less energy than you get out of it.
Much less.
No if the laws of physics mean something to you.
Sorry if I stepped on any toes. Math gets rather impersonal sometimes. As a side note, I really hope I can open some eyes here and perhaps direct amateur research toward less-futile ends.
Cheers,
-ElectroNut