We propose a new second-order accurate finite volume (FV) scheme for diffusion equations with discontinuous coefficients on unstructured meshes. This scheme is based on a novel least squares reconstruction with a compact neighbourhood that computes the cell-centered diffusion fluxes from the normal diffusion flux at cell faces. We propose two variants of the scheme rooted in the concept of “alpha damping” (AD), which are both linearly exact on arbitrary mesh topologies even when the tangential diffusive fluxes are discontinuous. The first variant, referred to as AD-G (Alpha Damping-Gradient), effects the discretisation of the normal derivative while the second, referred to as AD-F (Alpha Damping-Flux), involves discretisation of the normal fluxes. It is shown that harmonic averaging for diffusion coefficient at the faces is essential for the accuracy of the solution with the AD-G scheme but this condition is not necessary for the AD-F scheme. Numerical experiments demonstrate that the proposed FV schemes estimate the solution and fluxes to second and first order accuracy respectively for discontinuous diffusion problems on generic polygonal meshes. Studies also show that these schemes are discrete extremum preserving (DEP) and can be implemented with relative ease for diffusion problems in existing legacy codes. © 2021 Elsevier Ltd