  In this work, the cross-section of the reaction $\gamma p \rightarrow
V(J/\psi,\psi(2S)) p$ from the production threshold to medium energy is studied
and systematically analyzed within two gluon exchange model. The obtained
numerical results are in agreement with experimental data and other theoretical
predictions. Under the assumption of the scalar form factor of dipole form, the
value of proton mass radius is calculated as $0.55\pm 0.09$ fm and $ 0.77\pm
0.12$ fm from the fit to the predicted $J/\psi $ and $\psi(2S)$ differential
cross-section, respectively. Finally the average value of proton mass radius is
estimated to be $\sqrt{\left<R^2_m \right>}=0.67\pm 0.11$ fm. Moreover, one
find that extracting mass radius from the near-threshold differential
cross-section of heavy quarkoniums is always affected by large $|t|_{min}$.
These obtained results may provide important theoretical reference for
understanding of nucleon structure and future relevant experiments.
