[RedDevil]

Back in the days this subject came up a dozen of times, in no small manner spurred on by RustyLugNut, I tried to find any scientific backing for the idea that tiny amounts of hydrogen would indeed have a significant effect on engine efficiency on regular car engines.
But I could find none.

It is clear that a relatively large amount of hydrogen does have an effect, as it is consumed as a fuel and affects the burn speed and pinging resistance of the air/fuel mixture.
So that is a thing.
But it requires an additional supply of hydrogen. Electrolysis could never generate enough, due to its inefficiency, and so it would never offset the gains.

If a very low amount of hydrogen would have the same effect then the inefficiency of electrolysis would be less of a problem.
There are studies that show the effect exists even with trace amounts; like the one that IIRC used a purpose built block with a 30-1 compression rate and octane fuel on severe lean burn (not unlike Rusty's X-prize setup).
Under those conditions even seed values of hydrogen do have an effect.
But not in anything resembling a regular car engine using ordinary fuel.

Then the addition of water mist can give a few % improvement in the right conditions, I assume by reducing pumping loss and by adding volume to the exhaust gas, which is especially useful when the head is hot enough anyway.

Even if you do find efficiency gains from using a 'HHO' bubbler that may have more to do with the water that comes with the bubbles.