  Studies of $e^+e^- \to D_s^+ \overline{D}{}^{(*)0}K^-$ and the $P$-wave
charmed-strange mesons are performed based on an $e^+e^-$ collision data sample
corresponding to an integrated luminosity of 567 pb$^{-1}$ collected with the
BESIII detector at $\sqrt{s}= 4.600$ GeV. The processes of $e^+e^-\to D_s^+
\overline{D}{*}^{0} K^-$ and $D_s^+ \overline{D}{}^{0} K^-$ are observed for
the first time and are found to be dominated by the modes $D_s^+
D_{s1}(2536)^-$ and $D_s^+ D^*_{s2}(2573)^-$, respectively. The Born cross
sections are measured to be $\sigma^{B}(e^+e^-\to D_s^+ \overline{D}{*}^{0}
K^-) = (10.1\pm2.3\pm0.8) pb$ and $\sigma^{B}(e^+e^-\to D_s^+
\overline{D}{}^{0} K^-) = (19.4\pm2.3\pm1.6) pb$, and the products of Born
cross section and the decay branching fraction are measured to be
$\sigma^{B}(e^+e^-\to D_s^+D_{s1}(2536)^- +
c.c.)\cdot\mathcal{B}(D_{s1}(2536)^- \to \overline{D}{*}^{0} K^-) = (7.5 \pm
1.8 \pm 0.7) pb$ and $\sigma^{B}(e^+e^-\to D_s^+ D^*_{s2}(2573)^- +
c.c.)\cdot\mathcal{B}(D^*_{s2}(2573)^- \to \overline{D}{}^{0} K^-) = (19.7 \pm
2.9 \pm 2.0) pb$. For the $D_{s1}(2536)^-$ and $D^*_{s2}(2573)^-$ mesons, the
masses and widths are measured to be $M(D_{s1}(2536)^-) = (2537.7 \pm 0.5 \pm
3.1)~MeV/c^2,$ $ \Gamma(D_{s1}(2536)^-)) = (1.7\pm 1.2 \pm 0.6)~\rm MeV,$ and
$M(D^*_{s2}(2573)^-) = (2570.7\pm 2.0 \pm 1.7)~MeV/c^2,$
$\Gamma(D^*_{s2}(2573)^-) = (17.2 \pm 3.6 \pm 1.1)~\rm MeV.$ The spin-parity of
the $D^*_{s2}(2573)^-$ meson is determined to be $J^P=2^{+}$. In addition, the
process $e^+e^-\to D_s^+ \overline{D}{}^{(*)0} K^-$ are searched for using the
data samples taken at four (two) center-of-mass energies between 4.416 (4.527)
and 4.575 GeV, and upper limits at the $90\%$ confidence level on the cross
sections are determined.
