Let C be the normalization of an integral plane curve of degree d with delta-ordinary nodes or cusps as its singularities. If delta = 0, then Namba proved that there is no linear series g(d-2)1 and that every g(d-1)1 is cut out by a pencil of lines passing through a point on C. The main purpose of this paper is to generalize his result to the case delta > 0. A typical one is as follows: If d greater-than-or-equal-to-2(k + 1) and delta < kd - (k + 1)2 + 3 for some k > 0, then C has no linear series g(d-3)1. We also show that if d-greater-than-or-equal-to-2k + 3 and delta < kd - (k + 1)2 + 2, then each linear series g(d-2)1 on C is cut out by a pencil of lines. We have similar results for g(d-1)1 and g(2d-9)1. Furthermore, we also show that all of our theorems are sharp.