The multi-flow tube (MFT) method is mainly designed for two-dimensional (2-D) radiation hydrodynamic problems with multi-material, whose computational mesh is Lagrangian-Eulerian hybrid. Quality of the MFT quadrilateral mesh is usually higher than that of a general quadrilateral mesh. We consider a so-called nine-point diffusion scheme on this kind of meshes, in which the method of calculating vertex unknowns is improved by taking advantage of mesh features. Several methods for eliminating vertex unknowns are proposed by using harmonic averaging points or a gradient reconstruction method. Numerical experiments show that our methods obtain almost second order accuracy on MFT meshes.