The energy-dependent cross section for $e^{+}{e}^{-}\to \eta \psi \left(2S\right)$ is measured at 18 center of mass energies from 4.288 to 4.951 GeV using the BESIII detector. Using the same data samples, we also perform the first search for the reaction $e^{+}{e}^{-}\to \eta \tilde{X}\left(3872\right)$, but no evidence is found for the $\tilde{X}\left(3872\right)$ in the ${\pi }^{+}{\pi }^{-}J/\psi$ mass distribution. At each of the 18 center of mass energies, upper limits at the 90% confidence level on the cross section for $e^{+}{e}^{-}\to \eta \psi \left(2S\right)$ and on the product of the $e^{+}{e}^{-}\to \eta \tilde{X}\left(3872\right)$ cross section with the branching fraction of $\tilde{X}\left(3872\right)\to {\pi }^{+}{\pi }^{-}J/\psi$ are reported.