We extend Kawachi's invariant for reduced points on a normal suface to an invariant defined for possibly non-reduced zero-dimensional closed subschemes. (or fat points) on a surface with rational singularities. We give an upper bound of this invariant in terms of the length of the fat point and the discrepancy of the minimal resolution. As an application we prove a Reider-type theorem for k-very ampleness on surfaces with rational singularities. (C) 2001 Elsevier Science B.V. All rights reserved.