Abstract 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 $N_f$ and quark mass dependences induced by the $\mathbf{U}_A(1)$ anomaly dominance}. $M(q^2)$'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 $\Lambda_1={2.65(5)}~\mathrm{GeV}$ and $\Lambda_2={2.90(35)}~\mathrm{GeV}$. These results provide inputs for future theoretical and experimental studies on the Dalitz decays $J/\psi\to Pe^+e^-$. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Science and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.