Based on the perturbative quantum chromodynamics (pQCD) approach and the quasi-two-body approximation, we have studied the three-body decays $$B^{0}_{s}\rightarrow \psi (3686,3770)K \pi $$ $B_s^0\to \psi (3686,3770)K\pi$ , which include the contributions of the intermediate resonances $$\overline{K }^{*}_{0}(1430)^{0}$$ ${\overline{K}}_0^{\ast }{\left(1430\right)}^0$ , $$\overline{K }^{*}(892)^{0}$$ ${\overline{K}}^{\ast }{\left(892\right)}^0$ , $$\overline{K }^{*}(1410)^{0}$$ ${\overline{K}}^{\ast }{\left(1410\right)}^0$ , $$\overline{K }^{*}(1680)^{0}$$ ${\overline{K}}^{\ast }{\left(1680\right)}^0$ , and $$\overline{K }^{*}_{2}(1430)^{0}$$ ${\overline{K}}_2^{\ast }{\left(1430\right)}^0$ . The time-like form factors corresponding to the distribution amplitudes of the S-, P-, and D-wave of the kaon–pion pair were adopted in parameterized form, and describe the interactions between $$K $$ $K$ and $$\pi $$ $\pi$ in the resonance region. First, the decays $$B^{0}_{s}\rightarrow \psi (2S,1D)K ^{-}\pi ^{+}$$ $B_s^0\to \psi (2S,1D)K^{-}{\pi }^{+}$ were calculated, followed by the calculation of the branching ratios of the decays $$B^{0}_{s}\rightarrow \psi (3686,3770)K ^{-}\pi ^{+}$$ $B_s^0\to \psi (3686,3770)K^{-}{\pi }^{+}$ using the 2S–1D mixing scheme. In addition, the pQCD predictions for the decays $$B^{0}_{s}\rightarrow \psi (2S,1D)K \pi $$ $B_s^0\to \psi (2S,1D)K\pi$ and $$B^{0}_{s}\rightarrow \psi (3686,3770)K \pi $$ $B_s^0\to \psi (3686,3770)K\pi$ were obtained using the narrow-width approximation relation given by the Clebsch–Gordan coefficients. Our work shows that the $$\overline{K }^{*}(892)^{0}$$ ${\overline{K}}^{\ast }{\left(892\right)}^0$ resonance is the main contributor to the total decay, and the branching ratio and the longitudinal polarization fraction of the $$\psi (2S)\overline{K }^{*}(892)^{0}$$ $\psi \left(2S\right){\overline{K}}^{\ast }{\left(892\right)}^0$ decay mode agree well with the currently available data within errors. Furthermore, the theoretical predictions of the $$\psi (2S)$$ $\psi \left(2S\right)$ and $$\psi (3686)$$ $\psi \left(3686\right)$ 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 $$B^{0}_{s}\rightarrow \psi (3686,3770)K \pi $$ $B_s^0\to \psi (3686,3770)K\pi$ are of the order of $$10^{-5}$$ ${10}^{-5}$ and $$10^{-6}$$ ${10}^{-6}$ , respectively, which can be verified using the ongoing LHCb and Belle II experiments.