  Using data samples collected with the BESIII detector operating at the BEPCII
storage ring at center-of-mass energies from 4.178 to 4.600 GeV, we study the
process $e^+e^-\rightarrow\pi^{0}X(3872)\gamma$ and search for
$Z_c(4020)^{0}\rightarrow X(3872)\gamma$. We find no significant signal and set
upper limits on
$\sigma(e^+e^-\rightarrow\pi^{0}X(3872)\gamma)\cdot\mathcal{B}(X(3872)\rightarrow\pi^{+}\pi^{-}J/\psi)$
and
$\sigma(e^+e^-\rightarrow\pi^{0}Z_c(4020)^{0})\cdot\mathcal{B}(Z_c(4020)^{0}\rightarrow
X(3872)\gamma)\cdot\mathcal{B}(X(3872)\rightarrow\pi^{+}\pi^{-}J/\psi)$ for
each energy point at $90\%$ confidence level, which is of the order of several
tenths pb.
