Searches for $CP$ violation in the two-body decays $D_{(s)}^{+}rightarrow h^{+}pi^{0}$ and $D_{(s)}^{+} rightarrow h^{+}eta$ (where $h^{+}$ denotes a $pi^{+}$ or $K^{+}$ meson) are performed using $pp$ collision data collected by the LHCb experiment corresponding to either 9 fb$^{-1}$ or 6 fb$^{-1}$ of integrated luminosity. The $pi^{0}$ and $eta$ mesons are reconstructed using the $e^{+}e^{-}gamma$ final state, which can proceed as three-body decays $pi^{0}rightarrow e^{+}e^{-}gamma$ and $eta rightarrow e^{+}e^{-}gamma$, or via the two-body decays $pi^{0}rightarrow gammagamma$ and $etarightarrowgammagamma$ followed by a photon conversion. The measurements are made relative to the control modes $D_{(s)}^{+}rightarrow K_{S}^{0} h^{+}$ to cancel the production and detection asymmetries. The $CP$ asymmetries are measured to be begin{align} mathcal{A}_{CP}(D^{+}rightarrow pi^{+}pi^{0}) &= (-1.3 pm 0.9 pm 0.6 )%, end{align} begin{align} mathcal{A}_{CP}(D^{+}rightarrow K^{+}pi^{0}) &= (-3.2 pm 4.7 pm 2.1 )%, end{align} begin{align} mathcal{A}_{CP}(D^{+}rightarrow pi^{+}eta) &= (-0.2 pm 0.8 pm 0.4 )%, end{align} begin{align} mathcal{A}_{CP}(D^{+}rightarrow K^{+}eta) &= (-6 pm 10 pm 4 )%, end{align} begin{align} mathcal{A}_{CP}(D_{s}^{+}rightarrow K^{+}pi^{0}) &= (-0.8 pm 3.9 pm 1.2 )%, end{align} begin{align} mathcal{A}_{CP}(D_{s}^{+}rightarrow pi^{+}eta) &= (0.8 pm 0.7 pm 0.5 )%, end{align} begin{align} mathcal{A}_{CP}(D_{s}^{+}rightarrow K^{+}eta) &= (0.9 pm 3.7 pm 1.1 )%, end{align} where the first uncertainties are statistical and the second systematic. These results are consistent with no $CP$ violation and mostly constitute the most precise measurements of $mathcal{A}_{CP}$ in these decay modes to date.
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