Mass measurements of fission and projectile fragments, produced via U238 and Xe124 primary beams, have been performed with the multiple-reflection time-of-flight mass spectrometer (MR-TOF-MS) of the Fragment Separator (FRS) Ion Catcher with a mass resolving power (FWHM) of up to 410 000 and an uncertainty of down to 6×10-8. The nuclides were produced and separated in flight with the fragment separator FRS at 300 to 1000 MeV/u and thermalized in a cryogenic stopping cell. The data-analysis procedure was developed to determine with highest accuracy the mass values and the corresponding uncertainties for the most challenging conditions: down to a few events in a spectrum and overlapping distributions, which can be distinguished from a single peak only by a broader peak shape. With this procedure, the resolution of low-lying isomers is increased by a factor of up to 3 compared to standard data analysis. The ground-state masses of 31 short-lived nuclides of 15 different elements with half-lives of down to 17.9 ms and count rates as low as 11 events per nuclide were determined. This is the first direct mass measurement for seven nuclides. The excitation energies and the isomer-to-ground-state ratios of six isomeric states with excitation energies of as little as 280 keV were measured. For nuclides with known mass values, the average relative deviation from the literature values is (4.5±5.3)×10-8. The measured two-neutron separation energies and their slopes near and at the N=126 and Z=82 shell closures indicate a strong element-dependent binding energy of the first neutron above the closed proton shell Z=82. The experimental results deviate strongly from the theoretical predictions, especially for N=126 and N=127.