Are Helium 3 and 4 being produced by earth's core?
The paper you are referring to has major flaws in it. Claims that should not be in any respectable scientific journal such as:
...the demise of the georeactor is approaching. Within the present level of uncertainty, one cannot say precisely when georeactor demise will occur, whether in the next century...
At some point in time after the georeactor dies, there will be no geomagnetic field and life on Earth will never be the same.
etc., have no place in a scientific article that is to be taken seriously by anyone. This is scaremongering that belongs in tabloids. This was publish in PNAS which is a reputable venue, so I don't know how it got through. Was this even reviewed? No idea. But let's talk about the actual science presented in the paper.
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.