A new single-particle effective potential is proposed. This potential allows one to take into account quantum-electrodynamic (QED) corrections in relativistic calculations of many-electron ions and neutral atoms. In particular, it can be used in the Dirac-Fock (DF) method, the relativistic density functional, the multiconfiguration DF method, and the relativistic method of superposition of configurations. The potential is constructed without fitting parameters. Self-energy corrections have been calculated for a number of neutral alkali atoms and Li-like ions to check the quality of the nonlocal potential proposed. Comparison with the data in the literature on the QED corrections obtained in nonempirical calculations based on the use of QED perturbation theory is performed.