  We calculate the form factor $M(q^2)$ for the Dalitz decay $J/\psi\to
\gamma^*(q^2)\eta_{(N_f=1)}$ with $\eta_{(N_f)}$ being the SU($N_f$) flavor
singlet pseudoscalar meson. The difference among the partial widths
$\Gamma(J/\psi\to \gamma \eta_{(N_f)})$ at different $N_f$ can be attributed in
part to the $\mathbf{U}_A(1)$ anomaly that induces a $N_f$ scaling. $M(q^2)$'s
in $N_f=1,2$ are both well described by the single pole model
$M(q^2)=M(0)/(1-q^2/\Lambda^2)$. Combined with the known experimental results
of the Dalitz decays $J/\psi\to Pe^+e^-$, the pseudoscalar mass $m_P$
dependence of the pole parameter $\Lambda$ is approximated by
$\Lambda(m_P^2)=\Lambda_1(1-m_P^2/\Lambda_2^2)$ with
$\Lambda_1=2.64(4)~\mathrm{GeV}$ and $\Lambda_2=2.97(33)~\mathrm{GeV}$. These
results provide inputs for future theoretical and experimental studies on the
Dalitz decays $J/\psi\to Pe^+e^-$.
