  Born cross sections for the processes $e^+e^- \to \omega\eta$ and $e^+e^- \to
\omega\pi^{0}$ have been determined for center-of-mass energies between 2.00
and 3.08 GeV with the BESIII detector at the BEPCII collider. The results
obtained in this work are consistent with previous measurements but with
improved precision. Two resonant structures are observed. In the $e^{+}e^{-}
\to \omega\eta$ cross sections, a resonance with a mass of $(2179 \pm 21 \pm
3)\text{MeV}/c^2$ and a width of $(89 \pm 28 \pm 5)\text{MeV}$ is observed with
a significance of 6.1$\sigma$. Its properties are consistent with the
$\phi(2170)$. In the $e^{+}e^{-} \to\omega\pi^{0}$ cross sections, a resonance
denoted $Y(2040)$ is observed with a significance of more than 10$\sigma$. Its
mass and width are determined to be $(2034 \pm 13 \pm 9)\text{MeV}/c^2$ and
$(234 \pm 30 \pm 25)\text{MeV}$, respectively, where the first uncertainties
are statistical and the second ones are systematic.
