We present an upper bound for the total variation distance between the generalized polynomial distribution and a finite signed measure, which is the convolution of two finite signed measures; one of which is of Kornya-Presman type. In the one-dimensional Poisson case, such a finite signed measure was first considered by K. Borovkov and D. Pfeifer {[}J. Appl. Probab., 33 (1996), pp. 146-155]. We give asymptotic relations in the one-dimensional case, and, as an example, the independent identically distributed record model is investigated. It turns out that here the approximation is of order O(n(-s)(log n)(-(s+1)/2)) for s being a fixed positive integer, whereas in the approximation with simple Kornya-Presman signed measures, we only have the rate O((log n)(-(s+1)/2)).