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Direct mass measurements of neutron-deficient nuclides around the $N=50$ shell closure below $^{100}$Sn were performed at the FRS Ion Catcher (FRS-IC) at GSI, Germany. The nuclei were produced by projectile fragmentation of $^{124}$Xe, separated in the fragment separator FRS and delivered to the FRS-IC. The masses of 14 ground states and two isomers were measured with relative mass uncertainties down to $1\times 10^{-7}$ using the multiple-reflection time-of-flight mass spectrometer of the FRS-IC, including the first direct mass measurements of $^{98}$Cd and $^{97}$Rh. A new $Q_\mathrm{EC} = 5437\pm67$ keV was obtained for $^{98}$Cd, resulting in a summed Gamow-Teller (GT) strength for the five observed transitions ($0^+\longrightarrow1^+$) as $B(\text{GT})=2.94^{+0.32}_{-0.28}$. Investigation of this result in state-of-the-art shell model approaches sheds light into a better understanding of the GT transitions in even-even isotones at $N=50$. The excitation energy of the long-lived isomeric state in $^{94}$Rh was determined for the first time to be $293\pm 21$ keV. This, together with the shell model calculations, allows the level ordering in $^{94}$Rh to be understood.