We report a study of the processes of $e^+e^-to K^+ (D_s^- D^{*0} + D^{*-}_s D^0)$ based on $e^+e^-$ annihilation samples collected with the BESIII detector operating at BEPCII at five center-of-mass energies ranging from 4.628 to 4.698 GeV with a total integrated luminosity of 3.7 fb$^{-1}$. An excess over the known contributions of the conventional charmed mesons is observed near the $D_s^- D^{*0}$ and $D^{*-}_s D^0$ mass thresholds in the $K^{+}$ recoil-mass spectrum for events collected at $sqrt{s}=4.681$ GeV. The structure matches a mass-dependent-width Breit-Wigner line shape, whose pole mass and width are determined as $(3982.5^{+1.8}_{-2.6}pm2.1)$ MeV/$c^2$ and $(12.8^{+5.3}_{-4.4}pm3.0)$ MeV, respectively. The first uncertainties are statistical and the second are systematic. The significance of the resonance hypothesis is estimated to be 5.3 $sigma$ over the contributions only from the conventional charmed mesons. This is the first candidate of the charged hidden-charm tetraquark with strangeness, decaying into $D_s^- D^{*0}$ and $D^{*-}_s D^0$. However, the properties of the excess need further exploration with more statistics.
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