A lot of people says the same than you, and a lot of people say the opposite. Some say it is impossible getting more energy than what you put in.
However, when I burn gasoline, when I burn coal, when I burn fuelwood, or when I burn Plutonium, I get a lot more energy out of them than the energy I input into the equation to split these fuels out to burn or whatever the chemical reaction is.
It IS impossible to get more energy out of a fuel than what was put in. The thing is, you don't have to be the one who put all of the energy in. In the case of fossil fuels most of the energy comes from decayed organic matter that has been processed in high heat and pressure. The organic matter supplies the initial energy. Everything there after is refinement until you finally put it into your gas tank. A lot of energy has been wasted in the process and you cannot get anywhere near the energy output from the fuel as what was put into it over thousands/millions of years.
So, instead, I want to make a real test.
The first hydrogen generator I burned used 30 A and generated about 1,5 lt/min of HHO.
The one I am testing now, uses 15 A and produces 3 times more HHO.
What I mean, is similarly to any other issue, you may have different models with different efficiency.
Yes, you will see difference efficiencies depending on the process.
Just consider this. For your scheme to work you'd need to get more power out of burning the HHO gas than you put in. You are using the alternator to provide power. Where does the alternator get it's power from? The engine. Let's say that the alternator is 90% efficient. Now, thermal engines (which fuel powered engines are) are around 10-20% efficient. Multiply .20 times 0.90 and the total energy efficiency of fuel into electricity in the alternator is 18%. This means that for every 100 units worth of energy you burn you are getting 18 back to put into electricity. Electrolysis isn't 100% efficient either. By the time you get down to it you are looking at maybe a maximum energy conversion efficiency of 10%. This means you'd have to burn 10 gallons worth of water in order to extract enough energy to convert 1 gallon to HHO. That is NOT efficiency. Now, you can argue that you aren't converting all of the engine's work into power to split water. And you aren't converting all of the energy in that way. But, what energy you do use will have the efficiencies outlined. So, once again, If you consider gasoline and water to have the same energy potential per gallon (which they don't. Gasoline is one of the most dense energy forms available) then it would take 10 gallons of gas to be able to release and burn the energy from one gallon of water.
Some people argue that burning a little hydrogen along with the gasoline actually increases the efficiency of the engine. That could be true. But, as you can see above, it would have to increase the efficiency of the engine A LOT in order to even break even. Also, beware: burning hydrogen will increase the head temperature of your engine. If you burn too much you will warp your head and ruin the engine.
In summary: I will test it myself to be sure to say: "from the best of my experience, it does / doesn´t improve yield".
Sure, I don't mean to tell you NOT to do it. It's your choice. I'm just trying to let you know that what you really want to have happen is almost certainly physically impossible. The best way for you to learn that is to try it yourself. That will give you a much better understanding than anything I could ever say to you. In the end you'll end up more experienced in a variety of areas. That's not a bad thing at all. In fact, if I'm somehow wrong then you'll have done the world a favor by conducting detailed analysis.
Sorry for this long post. I don't mean to p$@% in your cornflakes but I do think that you should be aware of the potential roadblocks you are about to face.