  Many particles predicted by extensions of the Standard Model feature
interactions with neutrinos, e.g., Majoron-like bosons $\phi$. If the mass of
$\phi$ is larger than about 10 keV, they can be produced abundantly in the core
of the next galactic core-collapse supernova through neutrino coalescence, and
leave it with energies of around 100 MeV. Their subsequent decay to high-energy
neutrinos and anti-neutrinos provides a distinctive signature at Earth. Ongoing
and planned neutrino and dark matter experiments allow us to reconstruct the
energy, flavor, and time of arrival of these high-energy neutrinos. For the
first time, we show that these measurements can help pinpointing the mass of
$\phi$ and its couplings to neutrinos of different flavor. Our results can be
generalized in a straightforward manner to other hypothetical feebly
interacting particles, like novel gauge bosons or heavy neutral leptons, that
decay into neutrinos.
