I don't think that is the theory, at least from what I have been reading.
The HHO causes a more complete and clean combustion and actually lowers tail pipe emissions. The cleaner more fully combusted exhaust is read by the O2 as a lean condition having less un-burnt fuel.
"IF" this is the case, the ECM will adjust to make the mix more rich so the more fuel added cancels out the HHO benefit. The ones fooling the O2 are doing so to maintain the same A/F as before.
No, a cleaner combustion will cause less oxygen to be present in the exhaust. REmember, right now there is about 2% unburned fuel and 2% unburned oxygen, which then burn up in the catalyst, to make it work. The mixture rapidly fluctuates rich/lean/rich/lean to make the 3 way catalyst to function, and to get any useful reading out of a narrow band oxygen sensor.
So, a more complete combustion uses up more oxygen , and the computer will LEAN out the mixture (cut back on fuel) to restore the 2% oxygen that it expects to see. (It's an oxygen sensor, not a fuel sensor!) Incidentally, this means that a misfiring cylinder will produce more unburned fuel, and more unburned oxygen. The computer will then add even more fuel to reduce the amount of oxygen in the exhaust. Not an ideal situation. Newer vehicles will set a code, but old ones will keep going.
Had the manufacturers, with the epa's blessing, instead decided to use a CO sensor, or and HC sensor, this problem would not occur, but the oxygen sensor was chosen because it matches the functionality of the (inefficient) 3 way catalyst emissions scheme. A more efficient use would be to use lean combustion at cruise, which they do in Europe (and get better mpg and higher NOx emissions) and maybe a urea catalyst to reduce NOx in the tailpipe. So now you know that gassers may be going that direction too (urea), and direct injection, just like diesels.
There are many vaccuum leak devices, sold as economizers, that the EPA has tested. One such device, tested on an old 60's Plymouth, reduced both fuel consumption and emissions at cruise by about 16%, as acknowledged by the EPA itself. There is a site that lists devices tested by epa.