We propose an error analysis of a weak Galerkin method for second-order elliptic equations with minimal regularity requirements, i.e., the exact solution u belongs to H1+s(Ω) with s>0 possibly close to zero. In this case, the conormal derivative of u is not well-defined on the mesh faces, which results in a legitimacy problem of the error equation originally derived under smooth assumption. We employ the weak meaning of the conormal derivative introduced in [Ern, A., & Guermond, J. L. (2021). Quasi-optimal Nonconforming Approximation of Elliptic PDEs with Contrasted Coefficients andH1+s,s>0, Regularity. Foundations of Computational Mathematics, 1–36.] to establish new error equation and optimal a priori error estimates for low regular solutions on polytopal meshes. We also investigate the corresponding elliptic eigenvalue problems. The numerical experiments verify the theoretical results and agree with the benchmark eigenvalue problem.