Using data samples with an integrated luminosity of $4.5{\mathrm{fb}}^{-1}$ collected by the BESIII detector at center-of-mass energies ranging from 4.66 GeV to 4.95 GeV, we study the processes of $e^{+}{e}^{-}\to \omega X\left(3872\right)$ and $e^{+}{e}^{-}\to \gamma X\left(3872\right)$. With the $e^{+}{e}^{-}\to \omega X\left(3872\right)$ process, the branching fraction ratio $R\equiv \frac{B\left(X\right(3872)\to \gamma J/\psi )}{B\left(X\right(3872)\to {\pi }^{+}{\pi }^{-}J/\psi )}$ is measured to be $0.38\pm 0.20_{\mathrm{stat}}\pm 0.01_{\mathrm{syst}}$ ($R<0.83$ at 90% confidence level). In addition, we measure the ratio of the average cross section of $e^{+}{e}^{-}\to \omega X\left(3872\right)$ to $e^{+}{e}^{-}\to \omega {\chi }_{c1}\left(\omega {\chi }_{c2}\right)$ to be ${\sigma }_{\omega X\left(3872\right)}/{\sigma }_{\omega {\chi }_{c1}}({\sigma }_{\omega X\left(3872\right)}/{\sigma }_{\omega {\chi }_{c2}})=5.2\pm {1.0}_{\mathrm{stat}}\pm {1.9}_{\mathrm{syst}}(5.5\pm {1.1}_{\mathrm{stat}}\pm {2.4}_{\mathrm{syst}})$. Finally, we search for the process of $e^{+}{e}^{-}\to \gamma X\left(3872\right)$, and no obvious signal is observed. The upper limit on the ratio of the average cross section of $e^{+}{e}^{-}\to \gamma X\left(3872\right)$ to $e^{+}{e}^{-}\to \omega X\left(3872\right)$ is set as ${\sigma }_{\gamma X\left(3872\right)}/{\sigma }_{\omega X\left(3872\right)}<0.23$ at 90% confidence level.