We investigate properties of the ground-state and first radially excited four-quark mesons ${X}_{0}$ and ${X}_{0}^{\ensuremath{'}}$ with a diquark-antidiquark structure $[ud][\overline{c}\overline{s}]$ and spin-parities ${J}^{\mathrm{P}}={0}^{+}$. Our aim is to reveal whether or not one of these states can be identified with the resonance ${X}_{0}(2900)$, recently discovered by the LHCb Collaboration. We model ${X}_{0}$ and ${X}_{0}^{\ensuremath{'}}$ as tetraquarks composed of either axial-vector or scalar diquark and antidiquark pairs. Their spectroscopic parameters are computed by employing the QCD two-point sum rule method and including vacuum condensates up to dimension 15 in the analysis. For an axial-axial structure of ${X}_{0}^{(\ensuremath{'})}$, we find partial widths of the decays ${X}_{0}^{(\ensuremath{'})}\ensuremath{\rightarrow}{D}^{\ensuremath{-}}{K}^{+}$ and ${X}_{0}^{(\ensuremath{'})}\ensuremath{\rightarrow}{D}^{0}{K}^{0}$, and estimate full widths of the states ${X}_{0}^{(\ensuremath{'})}$. To this end, we calculate the strong couplings at the vertices ${X}_{0}^{(\ensuremath{'})}DK$ in the framework of the light-cone sum rule method. We also use technical approaches of the soft-meson approximation necessary to analyze tetraquark-meson-meson vertices. We obtain $m=(2545\ifmmode\pm\else\textpm\fi{}160)\text{ }\text{ }\mathrm{MeV}$ and ${m}^{\ensuremath{'}}=(3320\ifmmode\pm\else\textpm\fi{}120)\text{ }\text{ }\mathrm{MeV}$ [${m}_{\mathrm{S}}=(2663\ifmmode\pm\else\textpm\fi{}110)\text{ }\text{ }\mathrm{MeV}$ and ${m}_{\mathrm{S}}^{\ensuremath{'}}=(3325\ifmmode\pm\else\textpm\fi{}85)\text{ }\text{ }\mathrm{MeV}$ for a scalar-scalar current] for the masses of the particles ${X}_{0}$ and ${X}_{0}^{\ensuremath{'}}$, as well as estimates for their full widths ${\mathrm{\ensuremath{\Gamma}}}_{0}=(140\ifmmode\pm\else\textpm\fi{}29)\text{ }\text{ }\mathrm{MeV}$ and ${\mathrm{\ensuremath{\Gamma}}}_{0}^{\ensuremath{'}}=(110\ifmmode\pm\else\textpm\fi{}25)\text{ }\text{ }\mathrm{MeV}$, which allow us to interpret that neither is the resonance ${X}_{0}(2900)$. At the same time, these predictions provide important information about the ground-state and radially excited diquark-antidiquark structures ${X}_{0}$ and ${X}_{0}^{\ensuremath{'}}$, which should be objects of future experimental and theoretical studies.