  We report corrected results on a study of the electromagnetic Dalitz decay
$J/\psi \to e^+e^- \eta$ and search for di-electron decays of a light dark
gauge boson ($\gamma'$) in $J/\psi \to \gamma' \eta$ with the two $\eta$ decay
modes $\eta \rightarrow \gamma \gamma$ and $\eta \rightarrow \pi^+\pi^-\pi^0$,
using $(1310.6\pm 7.0)\times10^6$ $J/\psi$ events collected with the BESIII
detector. In particular we update the measurement of the pole mass of the
transition form factor (TFF) of $J/\psi \to e^+e^- \eta$ to be $\Lambda = 2.56
\pm 0.04({\rm stat}) \pm 0.03({\rm syst})$ GeV/$c^2$. The branching fraction of
$J/\psi \to e^+e^-\eta$ and the kinetic-mixing strength between the Standard
Model photon and $\gamma'$ are also updated after taking the corrected
$\Lambda$ value into account. The measured branching fraction of $J/\psi \to
e^+e^- \eta$ is updated to be $(1.42 \pm 0.04({\rm stat}) \pm 0.07 ({\rm
syst}))\times 10^{-5}$, where the first and second uncertainties are
statistical and systematic, respectively.
