Note this is a 2005 paper, so they are probably familiar with the debate. (Note I rely on these papers more than the Wiki entries, since Wiki doesn't really adress the composition of the core anyway.)
Some of this heat comes from the decay of radioactive elements. Based on studies of primitive meteorites known as carbonaceous chondrites, geologists have estimated Earth's uranium and thorium content and calculated that about 19 terawatts can be attributed to radioactivity. But until now there has been nothing definitive about exactly how much uranium there is in the planet, says geologist Bill McDonough of the University of Maryland in College Park. "There are fundamental uncertainties."
Now, the KamLAND antineutrino detector in Kamioka, Japan, has counted such antineutrinos. An international team of scientists analysed the data and found about 16.2 million antineutrinos per square centimetre per second streaming out from Earth's core. They calculate that the nuclear reactions creating these particles could be generating as much as 60 terawatts, but are most likely putting out about 24 terawatts (Nature, vol 436, p 499). "We have made the first measurements of the radioactivity of the whole of Earth," says John Learned, who heads the KamLAND group at the University of Hawaii in Manoa. The KamLAND group's finding is like unwrapping a birthday present, says McDonough.
Taking the most extreme possibility, a figure of 60 TW would indicate that possibly U is several times more abundant than previously believed, or that induced fission is ongoing, but necessarily involving a much smaller amount of material. I don't think the upper bound allows the possibility of a very large mass of U at the core -- maybe more than was postulated earlier, but not hugely more.
The fifth link I posted indicates there's some reason to suspect K, more than U, is responsible for any anomalously high readings.
We can't mine the Earth's core, but its heat and radiation signatures should give us a better idea of what's down there.
There may be uncertainties about the total amount of U present in the Earth, but not as much doubt about how much is accessibly near the surface. Finding a mass of U at the core helps nobody, in terms of meeting energy demands.
(PS: Herndon mentions a mass about 8km in radius. That would still be an extremely small quantity compared to O, Si, Al, Na, K, Ca, Fe, Al, Fe, Mg, Ti, by a factor of many thousands. That's a really far cry from being the second most abundant element.)