Measurement of the $e^{+}{e}^{-}\to \eta J\mathrm{/}\psi$ cross section and search for $e^{+}{e}^{-}\to {\pi }^{0}J\mathrm{/}\psi$ at center-of-mass energies between 3.810 and 4.600~GeV

Using data samples collected with the BESIII detector operating at the BEPCII collider at center-of-mass energies from 3.810 to 4.600 GeV, we perform a study of $e^{+}{e}^{-}\to \eta J\mathrm{/}\psi$ and ${\pi }^{0}J\mathrm{/}\psi$. Statistically significant signals of $e^{+}{e}^{-}\to \eta J\mathrm{/}\psi$ are observed at $\sqrt{s}$ = 4.190, 4.210, 4.220, 4.230, 4.245, 4.260, 4.360 and 4.420 GeV, while no signals of $e^{+}{e}^{-}\to {\pi }^{0}J\mathrm{/}\psi$ are observed. The measured energy-dependent Born cross section for $e^{+}{e}^{-}\to \eta J\mathrm{/}\psi$ shows an enhancement around 4.2~GeV. The measurement is compatible with an earlier measurement by Belle, but with a significantly improved precision.