  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 $\pi^{0}\rightarrow \gamma\gamma$ and
$\eta\rightarrow\gamma\gamma$ 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.
