Supernovae are powered by gravitational energy. Specifically, the change in gravitational potential energy when an electron-degenerate core, the mass of the Sun and radius of the Earth, collapsed rapidly ($<1$ s) to a radius of $\sim 10$ km.
Where does this energy go? Initially it is shared between escaping neutrinos and making the the dense core extremely hot ($10^{11}$ K). However, as the central core achieves neutron star densities, the collapse is abruptly halted by the repulsion between closely packed nucleons and the consequent "bounce" drives a shockwave outwards.
The shock wave is not powerful enough to power the supernova. Instead, the driving force at play is the increased opacity of the hot, dense inner regions to neutrinos.
A basic rule of thumb is it takes a light year of lead to stop a neutrino. But neutron star densities are about $10^{13}$ times that of lead, so the equivalent is around 10 km of neutron star material. It is the absorption of the energy and momentum of just a small fraction (a few percent) of the neutrinos produced in the hot, but rapidly cooling, core, that drives the supernova explosion and throws off the outer envelope of the star.
Now to your specific question:
The hot core produces neutrinos chiefly via the URCA process. These are cycles of beta decay and inverse beta decay. $$ n \rightarrow p + e + \bar{\nu}$$ $$ p + e \rightarrow n + \nu $$ At very high temperatures, these processes can come into thermal equilibrium (ordinarily, the second process is energetically unfavorable). The resultant (anti) neutrinos have a range of energies of order $k_B T \sim 10$ MeV and would be called "thermal neutrinos".
Electron and positrons produced by pair production at high energies can also annihilate to produce neutrino/anti neutrinos pairs. Again, the energies of the neutrinos reflect the thermal energies of the electrons and positrons. There is also a scattering interaction between nucleons that can produce neutrino/anti neutrino pairs called "neutrino bremsstrahlung".
These thermal neutrinos are a coolant. They take away energy from the core, allowing it to cool by an order of magnitude in seconds by emitting $\sim 10^{57}$ such neutrinos (the thermal energy of the core divided by 10 MeV). However, in these first few seconds after core collapse, the region just outside where these neutrinos are produced is able to absorb some of them, which drives the explosion.