  The decays $\psi_2(3823)\rightarrow\gamma\chi_{c0,1,2}, \pi^+\pi^-J/\psi,
\pi^0\pi^0J/\psi, \eta J/\psi$, and $\pi^0J/\psi$ are searched for using the
reaction $e^+e^-\rightarrow\pi^+\pi^-\psi_2(3823)$ in a 19 fb$^{-1}$ data
sample collected at center-of-mass energies between 4.1 and 4.7 GeV with the
BESIII detector. The process $\psi_2(3823)\rightarrow\gamma\chi_{c1}$ is
observed in a 9 fb$^{-1}$ data sample in the center-of-mass energy range 4.3 to
4.7 GeV, which confirms a previous observation but with a higher significance
of $11.8\sigma$, and evidence for $\psi_2(3823)\rightarrow\gamma\chi_{c2}$ is
found with a significance of $3.2\sigma$ for the first time. The
branching-fraction ratio
$\frac{\mathcal{B}(\psi_2(3823)\rightarrow\gamma\chi_{c2})}{\mathcal{B}(\psi_2(3823)\rightarrow\gamma\chi_{c1})}$
is determined. No significant $\psi_2(3823)$ signals are observed for any of
the other decay channels. Upper limits of branching-fraction ratios for
$\psi_2(3823)\rightarrow\pi^+\pi^-J/\psi, \pi^0\pi^0J/\psi, \eta J/\psi,
\pi^0J/\psi, \gamma\chi_{c0}$ relative to
$\psi_2(3823)\rightarrow\gamma\chi_{c1}$ are reported. The process
$e^+e^-\rightarrow\pi^0\pi^0\psi_2(3823)$ is also searched for, and we find
evidence for the process with a significance of $4.3\sigma$. The average
cross-section ratio
$\frac{\sigma(e^+e^-\rightarrow\pi^0\pi^0\psi_2(3823))}{\sigma(e^+e^-\rightarrow\pi^+\pi^-\psi_2(3823))}$
is also determined.
