In this series of papers about partial Gaussian sums the author is concerned with the estimation of sums of the form \[ S_ a(N,H):=\sum^{N+H}_{n=N+1}\chi(n)e^{2\pi ina/k},\qquad a,N,H\in\mathbb{Z},\;H>0, \] where \(\chi\) denotes a non-principal Dirichlet character modulo \(k\). In part I [Bull. Lond. Math. Soc. 20, 589-592 (1988; Zbl 0667.10024)] he showed that, for any integer \(r\geq 2\) and any prime \(k\), \[ S_ a(N,H)\ll H^{1-1/r} k^{1/4(r-1)}\log^ 2 k. \] Part II of these investigations [Bull. Lond. Math. Soc. 21, 153-158 (1989; Zbl 0667.10025)] generalizes this estimate (in the case \(r=3\)) to arbitrary moduli \(k\). In the present paper the author succeeds in extending his first result (in the case \(r=4\)) to prime power moduli \(k=p^ \alpha\), \(p>3\).