We derive new constraints on combination of dark matter-electron cross section (${\ensuremath{\sigma}}_{\ensuremath{\chi}e}$) and dark matter-neutrino cross section (${\ensuremath{\sigma}}_{\ensuremath{\chi}\ensuremath{\nu}}$) utilizing the gain in kinetic energy of the dark matter (DM) particles due to scattering with the cosmic ray electrons and the diffuse supernova neutrino background (DSNB). Since the flux of the DSNB neutrinos is comparable to the cosmic ray (CR) electron flux in the energy range $\ensuremath{\sim}1\text{ }\text{ }\mathrm{MeV}--50\text{ }\text{ }\mathrm{MeV}$, scattering with the DSNB neutrinos can also boost low-mass DM significantly in addition to the boost due to interaction with the CR electrons. We use the XENON1T as well as the Super-Kamiokande data to derive bounds on ${\ensuremath{\sigma}}_{\ensuremath{\chi}e}$ and ${\ensuremath{\sigma}}_{\ensuremath{\chi}\ensuremath{\nu}}$. While our bounds for ${\ensuremath{\sigma}}_{\ensuremath{\chi}e}$ are comparable with those in the literature, we show that the Super-Kamiokande experiment provides the strongest constraint on ${\ensuremath{\sigma}}_{\ensuremath{\chi}\ensuremath{\nu}}$ for DM masses below a few MeV.
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