<p>Based on the Skyrme energy density functional and reaction <i>Q</i>-value, this study proposed an effective nucleus-nucleus potential for describing the capture barrier in heavy-ion fusion processes. The 443 extracted barrier heights were well reproduced with a root-mean-square (RMS) error of 1.53 MeV, and the RMS deviations with respect to 144 time-dependent Hartree-Fock capture barrier heights were only 1.05 MeV. Coupled with the Siwek-Wilczyński formula, wherein three parameters were determined by the proposed effective potentials, the measured capture cross sections at energies around the barriers were reasonably well reproduced for several fusion reactions induced by nearly spherical nuclei as well as by nuclei with large deformations, such as ¹⁵⁴Sm and ²³⁸U. The shallow capture pockets and small values of the average barrier radii resulted in the reduction of the capture cross sections for ^52,54Cr- and ⁶⁴ Ni-induced reactions, which were related to the synthesis of new super-heavy nuclei.</p>