A class of pseudoconvex domains with weak constraints on the boundary is introduced and for their Bergman kernels one obtains the principal term of the asymptotic expansion near the diagonal. The strictly pseudo-convex domains with C⁴-boundaries belong to the indicated class; for them one has obtained C. Fefferman's principal term for domains with C^∞-boundaries and the error estimate due to L. Boutet de Monvel and J. Sjöstrand.