The paper deals with complete Dirichlet character sums of type \[ S_ k(\chi,q)=\sum^ q_{\ell=1}{}' \chi(a\ell^ k+b) \qquad\text{or}\qquad T_ k(\chi,q)=\sum^ q_{\ell=1}{}' \chi((a\ell^ k+b)/(c\ell^ k+d)), \] where \(\chi\) denotes a primitive character \(\bmod q\). The dash indicates that the sums are restricted to those \(\ell\) for which \(c\ell^ k+d\not\equiv 0\bmod q\) and \(\chi(a\ell^ k+b)\), \(\chi((a\ell^ k+b)/(c\ell^ k+d))\not\equiv\text{const}\). The author calculates \(S_ k(\chi,q)\) and \(T_ k(\chi,q)\) for \(q=p^ n\), where \(p\) is a prime number and \(n\geq 2\). The results are similar to those known for generalized Gaussian sums. They can be used to obtain best possible lower bounds for complete multiple character sums involving certain polynomials or rational functions in several variables.