  We report on an analysis of the decay
$J/\psi\rightarrow\gamma\pi^0\eta^{\prime}$ using a sample of $(1310.6 \pm 7.0)
\times ~10^6~ J/\psi$ events collected with the BESIII detector. We search for
the CP-violating process $\eta_c\rightarrow\pi^0\eta^{\prime}$ and a dark gauge
boson $U'$ in $J/\psi\rightarrow
U^{\prime}\eta^{\prime},~U^{\prime}\rightarrow\gamma\pi^0,~\pi^0\rightarrow\gamma\gamma$.
No evidence of an $\eta_{c}$ signal is observed in the $\pi^0\eta^{\prime}$
invariant-mass spectrum and the upper limit of the branching fraction is
determined to be $7.2 \times~10^{-5}$ at the 90\% confidence level. We also
find no evidence of $U'$ production and set upper limits at the 90\% confidence
level on the product branching fraction $\mathcal{B}(J/\psi \rightarrow
U^{\prime} \eta^{\prime}) \times \mathcal{B}(U^{\prime} \rightarrow \pi^{0}
\gamma)$ in the range between $(0.8 - 6.5)\times 10^{-7}$ for 0.2 $\leq
m_{U^{\prime}} \leq 2.1 $GeV$/c^{2}$. In addition, we study the process
$J/\psi\rightarrow\omega\eta^{\prime}$ with $\omega\rightarrow\gamma\pi^0$. The
branching fraction of $J/\psi\rightarrow\omega\eta^{\prime}$ is found to be
$(1.87\pm0.09\pm0.12) \times 10^{-4}$, where the first uncertainty is
statistical and the second is systematic, with a precision that is improved by
a factor of 1.4 over the previously published BESIII measurement.
