  Considering the constraints from the flavor physics, precision electroweak
measurements, Higgs data and dark matter detections, we scan over the parameter
space of the MSSM and calculate the cross section of $e^+e^- \to h \gamma$ in
the allowed parameter space. Since the loop-induced gauge couplings
$h\gamma\gamma$ and $hZ\gamma$ can simultaneously contribute to the process
$e^+e^- \to h \gamma$, we find the cross section can be sizably enhanced by a
light stau, maximally 1.47(1.38) times larger than the SM prediction at
$\sqrt{s}=240(350)$ GeV. So with the high luminosity, the measurement of
$e^+e^- \to h \gamma$ may be used to test the anomalous gauge couplings
$h\gamma\gamma$ and $hZ\gamma$ in the MSSM at a Higgs Factory.
