A search for dark matter (DM) with mass in the sub-GeV region (0.32-1 GeV) was conducted by looking for an annual modulation signal in XMASS, a single-phase liquid xenon detector. Inelastic nuclear scattering accompanied by bremsstrahlung emission was used to search down to an electron equivalent energy of 1 keV. The data used had a live time of 2.8 years (3.5 years in calendar time), resulting in a total exposure of 2.38 ton-years. No significant modulation signal was observed and 90% confidence level upper limits of $1.6 times 10^{-33}$ cm$^2$ at 0.5 GeV was set for the DM-nucleon cross section. This is the first experimental result of a search for DM mediated by the bremsstrahlung effect. In addition, a search for DM with mass in the multi-GeV region (4-20 GeV) was conducted with a lower energy threshold than previous analysis of XMASS. Elastic nuclear scattering was used to search down to a nuclear recoil equivalent energy of 2.3 keV, and upper limits of 2.9 $times$10$^{-42}$ cm$^2$ at 8 GeV was obtained.
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