You're just twisting words into knots, let's use numbers.
1. Engine produces 15W of mechanical power
2. Alternator converts that to 14W of electricity
3. HHO generator converts that to 14W of HHO
4. Engine burns that to produce 15W of mechanical power
Step 2 assumes 93% efficient alternator, but you can have 99.99% if you want.
In step 3 I generously allowed a 100% efficient HHO generator.
Step 4 requires >100% efficiency.
and the HHO already has 3 times the power
but it only takes 1 times the electrical input.
The first line is correct, but you just invented the 2nd line. It directly contradicts the HHO generator efficiencies that we sorted out on long ago.
How does the HHO generator "know" that it's making HHO instead of petrol? It can't even make petrol. Why should it s performance in any way depend on petrol's energy density? Will it mysteriously get even better if we imagine how much energy a kg of plutonium has?