I have a theory about why some people experience an increase in fuel efficiency when feeding HHO into the intake of their engines... the HHO isn't... at least, it's not all HHO. It's actually a mix of HHO and vapor carrying hydroxide (OH^-) radicals into the engine.

The two main processes during combustion are below. You'll note that water (and specifically the hydroxide radical derived from water) is essential for combustion to even take place:

This process occurs at higher temperatures:
OH^- + H ==> H₂O
H₂O + O ==> H₂O₂
H₂O₂ ==> OH^- + OH^-

This process dominates at lower temperatures, and competes with the above process at higher temperatures:
OO + H ==> HOO
HOO + H ==> H₂O₂
H₂O₂ ==> OH^- + OH^-

The above two mechanisms are very active at stripping hydrogen from the hydrocarbon fuel.

Now consider that we're not just burning hydrogen in the engine... if the hydrogen combustion process finishes fast enough (ie: before the exhaust valve opens), we get to burn CO to CO₂:

The above two processes strip the hydrogen from the hydrocarbon fuel, leaving behind the carbon, which combines with oxygen to form carbon monoxide:
C + O ==> CO

Because the two processes above are competing with the C => CO => CO₂ process for oxygen, and they're faster, the only method really left to convert CO to CO₂ is via hydroxides.

CO + OH^- ==> CO₂ + H
H + OH^- ==> H₂0
H₂O + O ==> H₂O₂
H₂O₂ ==> OH^- + OH^-
Go to top and repeat.

As you can see, the hydroxide radical plays an important role in converting carbon monoxide to carbon dioxide, as well. And that process adds additional heat which can be converted to additional work.

So, my theory is that the HHO fed into the engine aids rapid completion of hydrocarbon combustion such that the CO => CO₂ process can take place before the exhaust valve opens, adding additional heat to the cylinder, which is extracted as work.

Of course, there's no need to expend all the electricity to generate the HHO... putting a voltage across a container of water such that the voltage doesn't exceed the water's ionization constant merely divides the water into OH^- (hydroxide) and H₃O (hydronium) (ie: the low pH and high pH constituents of water).

So putting just enough voltage across the water to cause that separation of the water into its high and low pH components (but not so high a voltage that you exceed the ionization constant of the water by much, thus keeping current to negligible levels, and not splitting the water into hydrogen and oxygen), then pumping from the OH^- side of your water tank and injecting that hydroxide rich water into your engine will jump-start the combustion process (because the hydroxide already exists, it doesn't need to be split from water in-cylinder, thereby skipping two steps as outlined above and immediately commencing with stripping hydrogen from the hydrocarbon fuel and burning it), thus allowing the hydrocarbon burn to complete faster, thus allowing the CO => CO₂ burn to initiate faster, thus adding to cylinder heat, thus increasing efficiency.