Mass measurements of fission and projectile fragments, produced via $^{238}$U and $^{124}$Xe primary beams, have been performed with the multiple-reflection time-of-flight mass spectrometer (MR-TOF-MS) of the FRS Ion Catcher with a mass resolving powers (FWHM) up to 410,000 and an uncertainty of $6cdot 10^{-8}$. 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, characterized only by a broader common peak shape. With this procedure, the resolution of low-lying isomers is increased by a factor of up to three compared to standard data analysis. The ground-state masses of 31 short-lived nuclides of 15 different elements with half-lives 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 down to about 280~keV were measured. For nuclides with known mass values, the average relative deviation from the literature values is $(2.9 pm 6.2) cdot 10^{-8}$. 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.
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