The authors derive upper bounds for mixed exponential sums of type \[ S( \chi, a x^n + bx, p^m) = \sum_{_{\substack{ x=1\\ p \nmid x}}}^{p^m} \chi(x) e_{p^m}(a x^n + bx), \] where \(p^m\) is a prime power with \(m \geq 2\), \(a,b\) are integers, \(n \geq 2\), \(\chi\) is a multiplicative character mod \(p^m\) and \(e_{p^m}(x) = \exp(2 \pi i x/p^m)\) . If \(\chi\) is primitive or \(p \nmid (a,b)\) , then they obtain \[ |S(\chi, ax^n +bx, p^m)|\leq 2 n p^{2m/3}. \] If \(\chi\) is of conductor \(p\) with \(p \nmid (a,b)\), then \[ |S(\chi, ax^n +bx, p^m)|\leq n p^{m/2}. \]