  Based on the perturbative quantum chromodynamics (pQCD) approach and the
quasi-two-body approximation, we have studied the three-body decays
B^{0}_{s}\rightarrow \psi(3686,3770)\emph{K}\pi, which include the
contributions of the intermediate resonances \bar{\emph{K}}^{*}_{0}(1430)^{0},
\bar{\emph{K}}^{*}(892)^{0}, \bar{\emph{K}}^{*}(1410)^{0},
\bar{\emph{K}}^{*}(1680)^{0}, and \bar{\emph{K}}^{*}_{2}(1430)^{0}. The
time-like form factors corresponding to the distribution amplitudes of the S,
P, and D-wave of the kaon-pion pair have been adopted in the parameterized
form, which describe the interactions between \emph{K} and \pi in the resonance
region. First, the decays B^{0}_{s}\rightarrow \psi(2S,1D)\emph{K}^{-}\pi^{+}
have been calculated followed by the calculation of the branching ratios of the
decays B^{0}_{s}\rightarrow \psi(3686,3770)\emph{K}^{-}\pi^{+} using the 2S-1D
mixing scheme. In addition, the pQCD predictions for the decays
B^{0}_{s}\rightarrow \psi(2S,1D)\emph{K}\pi and B^{0}_{s}\rightarrow
\psi(3686,3770)\emph{K}\pi have been obtained using the narrow-width
approximation relation given by the Clebsch-Gorden coefficients. Our work shows
that the \bar{\emph{K}}^{*}(892)^{0} resonance is the main contributor to the
total decay, and the branching ratio and the longitudinal polarization fraction
of the \psi(2S)\bar{\emph{K}}^{*}(892)^{0} decay mode agree well with the
currently available data within errors. Furthermore, the theoretical
predictions of the \psi(2S) and \psi(3686) decay modes are very close,
indicating that they can be regarded as the same meson state. Finally, the pQCD
predictions for branching ratios of decays B^{0}_{s}\rightarrow
\psi(3686,3770)\emph{K}\pi are of the order of 10^{-5} and 10^{-6},
respectively, which can be verified using the ongoing LHCb and Belle II
experiments.
