Can hydrogen run a generator to produce hydrogen perpetually?

No. We all can't be versed on every aspect of our world but some parts are important to keep us from falling into poor investments and scams. There are several laws of thermodynamics, essentially how energy works in our world and are a fundamental part of our understanding of physics.

One of those laws suggests that energy like arithmetic will not change unless something operates to change the numbers. The energy that is put into a system (2 + 2) is always going to equal the energy that goes out of a system (4) So just as 2 + 2 is not going to suddenly equal 5 or 6 the energy we put into our system is not somehow going to create more energy. The best we can hope for is to create a system that will somehow gather more energy from another source like solar panels. In this case it is like saying 2 + 2 + 1 equals something more.

In your suggestion you want to add hydrogen to a generator and then use the resulting electricity to produce hydrogen by splitting water and then use that hydrogen to run the engine. But you are leaving out some of the elements. The engine is probably no more than about 25% efficient. That means that 75% of the energy in the hydrogen fuel you use is used to create heat and is lost to friction. So far your equation then looks like 2 + 2 - 3 . The generator also loses some energy to friction. Add to this it takes more energy to split water into hydrogen and oxygen than we recover when hydrogen and oxygen combine to produce some excess energy. So in every step along the way we are losing our desired goal to losses in energy like heat and friction.

But even if the system were somehow perfect it could not be any more perfect than to lose no energy. You will not be creating new new energy with such a system. Even if we could get it to run forever the system is not going to give us new energy we could use for something else as every bit would be used to run the system. This is why we say that efficiency can never be greater than 1 or 100%. And we can not have "over unity" or "perpetual motion" machines.

If we want "free energy" what we could do is make solar panels. And then use those solar panels to gather additional energy from a source outside the system and then produce hydrogen from solar energy. What we can't do is use a solar panel to produce electricity for a light that shines on the solar panel as once again this would be a closed system with no energy flowing into the system.

When it comes to energy systems look for the source of the energy. If it is supposed to be coming from inside the system perpetually then it is likely a scam. If there is an outside source it may be useful depending upon the economics and how much energy we get relative to our energy investment.