<p>In this paper, we present a locally conservative numerical method to solve the general elliptic problems on polygonal or polyhedral meshes. The approximation space of the method is based on enriching that of the virtual element method with elementwise constant functions. For the formulation of the new method, we use the thoughts of the virtual element method to modify the schemes of the enriched Galerkin method. The proposed method has a slightly larger number of degrees of freedom than that of the virtual element method but can preserve the locally conservative property and deal with complicated element geometries. We show the solvability and local conservation of the method and prove the optimal convergence rate of numerical solutions. We can implement the new method only with some slight modifications on the existing code. We also provide some numerical examples to illustrate the optimal convergence rate of the new method and advantages of this method than discontinuous Galerkin method and virtual element method.</p>