I'm not an expert on nuclear reactions so I can't comment on that actual model presented in the paper, but I can comment on the assumptions that are simply wrong. His model assumes a certain volume of uranium that exists in the core. Such a "block" of uranium doesn't exist. Uranium doesn't go in the core. He says:
The background as to why a large portion of the Earth's reservoir of uranium is expected to exist in the core, precipitate, and ultimately collect at the center of the Earth has been set forth in refs. 8–11...
But refs 8–10 are actually his own papers. So he made up a theory that uranium goes in the core, completely ignoring years of published experimental work by different groups saying that uranium does not go into any significant amounts in the core, and then bases his story on that. This is wrong. Some related questions:
Why is uranium only in the crust, really?
What percent of the Earth's core is uranium?
Also note the lack of neutrinos, as mentioned by David Hammen in one of the answers, something that should happen if this was true.
So your questions:
Is Herndon correct...
No, he is not.
If not, what is their origin?
Mantle sources with different 3He/4He ratios. There are many reasons for that. Mid ocean ridge basalts are mostly generated by melting of depleted mantle, that is, mantle that was already melted (at least) once. So it will not have too much of primordial 3He in it, and the ratio will be low. Hotspots such as Iceland or Hawaii tap a deeper mantle source that still has 3He in it, so the ratio will be higher. There are other factors that can modify the ratio. Episodes of depletion and recycling, metasomatism (by aqueous fluids or carbonatites, each with their own capability to transport U and Th), and some more than I can't think of at the moment. Anyway, there's no nuclear reactor in the core of the earth and we're not going to die due to its demise because it doesn't exist.