Can the Sun contain degenerate matter?
The answer to whether a normal star can contain a core of degenerate neutrons is a yes. Thorne &amp; Zytkow 1977 produced numerically models where a neutron star becomes embedded in the center of a massive giant or supergiant star, surrounded by a large gaseous envelope. In this scenario, the main source of energy becomes not nuclear fusion but instead gravitational contraction from matter flowing from the inner envelope onto the outer core. The energy production ratios are
$$L_{\text{nuc}}/L\approx0.04,\quad L_{\text{grav}}/L=0.96$$
for stars with $M_{\text{tot}}\leq10M_{\odot}$, as above this, convective envelopes form. The models, regardless of mass, predict some degree of shell burning, with hydrogen-, helium-, and carbon- burning layers outside the core. In the inner regions, temperatures are orders of magnitude higher than that required for the CNO cycle, which sets in at around 15 million Kelvin, actually (not 20 million K) and dominates over the p-p chain at about 17 million K.

Thorne &amp; Zytkow did find that for $M_{\text{tot}}&lt;2M_{\odot}$, the envelopes were unstable against radial adiabatic pulsations, implying that it's likely not possible to extend the analysis to the case of the Sun - at that mass, the object is quite vulnerable to instabilities. That said, I am fairly confident that we would be able to tell if there was a degenerate object in the solar core; unlike the red giant or supergiant case, there is no large envelope to hide it, and the mass of the neutron star would at least be comparable to or (much more likely) exceed the mass of the Sun itself.



Some points on the stability of neutron degenerate matter: The problem is actually with small quantities, not large quantities. Small amounts are incapable of remaining bound by their own gravity; the pressures involved are simply too high, and the solar core is nowhere near high-pressure enough to maintain stability. Optimistically, you'd need somewhere in the $0.1M_{\odot}\text{-}0.2M_{\odot}$ range at minimum for stability, though I would be surprised if a mass of degenerate matter this low was produced naturally - a neutron star of a more typical mass would likely have to lose mass somehow.