First measurement of the cross sections for $e^{+}{e}^{-}\to K^0{K}^{-}{\pi }^{+}J\mathrm{/}\psi +c\mathrm{.}c\mathrm{.}$ at $\sqrt{s}$ from 4.396 to 4.951 GeV

Using $e^{+}{e}^{-}$ collision data at 19 center-of-mass energies ranging from $4.396$ to $4.951\mathrm{G}\mathrm{e}\mathrm{V}$ corresponding to a total integrated luminosity of $8.86{\mathrm{f}\mathrm{b}}^{-1}$ collected by the BESIII detector, the process $e^{+}{e}^{-}\to K^0{K}^{-}{\pi }^{+}J\mathrm{/}\psi +c\mathrm{.}c\mathrm{.}$ is observed for the first time, with a statistical significance of $9.4\sigma$ summing up all the data samples. For this process, the cross section and the upper limit at the $90\mathrm{\%}$ confidence level are reported at each of the 19 center-of-mass energies.~No statistically significant vector structures are observed in the cross section line shape, nor are any intermediate states of $K\pi$, $K\bar{K}$, $K\bar{K}\pi$, $KJ\mathrm{/}\psi$, $\pi J\mathrm{/}\psi$, and $K\pi J\mathrm{/}\psi$ seen at individual energy points or in the combined data sample.