  The processes $X(3872)\to D^{*0}\bar{D^{0}}+c.c.,~\gamma J/\psi,~\gamma
\psi(2S),$ and $\gamma D^{+}D^{-}$ are searched for in a $9.0~\rm fb^{-1}$ data
sample collected at center-of-mass energies between $4.178$ and $4.278$ GeV
with the BESIII detector. We observe $X(3872)\to D^{*0}\bar{D^{0}}+c.c.$ and
find evidence for $X(3872)\to\gamma J/\psi$ with statistical significances of
$7.4\sigma$ and $3.5\sigma$, respectively. No evident signals for
$X(3872)\to\gamma\psi(2S)$ and $\gamma D^{+}D^{-}$ are found, and upper limit
on the relative branching ratio $R_{\gamma \psi}
\equiv\frac{\mathcal{B}(X(3872)\to\gamma\psi(2S))}{\mathcal{B}(X(3872)\to\gamma
J/\psi)}<0.59$ is set at 90$\%$ confidence level. Measurements of branching
ratios relative to decay $X(3872)\to\pi^+\pi^- J/\psi$ are also reported for
decays $X(3872)\to D^{*0}\bar{D^{0}}+c.c., ~\gamma\psi(2S),~\gamma J/\psi$,
$\gamma D^{+}D^{-}$, as well as the non-$D^{*0}\bar{D}^{0}$ three-body decays
$\pi^0 D^{0}\bar{D}^{0}$ and $\gamma D^{0}\bar{D}^{0}$.
