  We propose a leptoquark model with two scalar leptoquarks $S^{}_1 \left(
\bar{3},1,\frac{1}{3} \right)$ and $\widetilde{R}^{}_2 \left(3,2,\frac{1}{6}
\right)$ to give a combined explanation of neutrino masses, lepton flavor
mixing and the anomaly of muon $g-2$, satisfying the constraints from the
radiative decays of charged leptons. The neutrino masses are generated via
one-loop corrections resulting from a mixing between $S^{}_1$ and
$\widetilde{R}^{}_2$. With a set of specific textures for the leptoquark Yukawa
coupling matrices, the neutrino mass matrix possesses an approximate
$\mu$-$\tau$ reflection symmetry with $\left( M^{}_\nu \right)^{}_{ee} = 0$
only in favor of the normal neutrino mass ordering. We show that this model can
successfully explain the anomaly of muon $g-2$ and current experimental
neutrino oscillation data under the constraints from the radiative decays of
charged leptons.
