  Using data samples with an integrated luminosity of $4.5~\text{fb}^{-1}$
collected by the BESIII detector at center-of-mass energies ranging from 4.66
to 4.95 GeV, we study the processes of $e^+e^-\to\omega X(3872)$ and
$e^+e^-\to\gamma X(3872)$. With the $e^+e^-\to\omega X(3872)$ process, the
branching fraction ratio $R\equiv\frac{\mathcal{B}(X(3872)\to\gamma
J/\psi)}{\mathcal{B}(X(3872)\to\pi^+\pi^- J/\psi)}$ is measured to be
$0.38\pm0.20_\text{stat.}\pm0.01_\text{syst.}$ ($R< 0.83$ at 90\% confidence
level). In addition, we measure the ratio of the average cross section of
$e^+e^-\to\omega X(3872)$ to $e^+e^-\to\omega \chi_{c1}(\omega\chi_{c2})$ to be
$\sigma_{\omega X(3872)}/\sigma_{\omega\chi_{c1}}~(\sigma_{\omega
X(3872)}/\sigma_{\omega\chi_{c2}})=5.2\pm1.0_\text{stat.}\pm1.9_\text{syst.}~
(5.5\pm1.1_\text{stat.}\pm2.4_\text{syst.})$. Finally, we search for the
process of $e^+e^-\to\gamma X(3872)$, and no obvious signal is observed. The
upper limit on the ratio of the average cross section of $e^+e^-\to\gamma
X(3872)$ to $e^+e^-\to\omega X(3872)$ is set as $\sigma_{\gamma
X(3872)}/\sigma_{\omega X(3872)}<0.23$ at 90\% confidence level.
