In this paper we present a nonlinear finite volume scheme preserving positivity for heat conduction equations. The scheme uses both cell-centered and cell-vertex unknowns. The cell-vertex unknowns are treated as auxiliary ones and are eliminated by our newly developed second-order explicit interpolation formula on generalized polyhedral meshes. With the help of the additional parameters, it is not necessary to choose the stencil adaptively to obtain the convex decomposition of the co-normal vector and also is not required to replace the interpolation formula with positivity-preserving but usually low-order accurate ones whenever negative interpolated auxiliary unknowns appear. Moreover, the new flux approximation has a fixed stencil. These features make our scheme more efficient compared with other existing methods based on Le Potier's nonlinear two-point approximation, especially in 3D. Numerical experiments show that the scheme maintains the positivity of the continuous solution and has nearly second-order accuracy for the solution on the distorted meshes where the diffusion tensor may be anisotropic and discontinuous.