Let \(g\) b the smallest positive integer such that every element of \(A = \mathbb{Z}/ p^n \mathbb{Z}\) is of the form \(\sum_{j=1}^s a_j x_j^k\) for some \(x_j \in A\), given \(a_j \in A\) and given \(k\). The \(x_j\)’s are required also to be outside the range of the canonical map \(A \mapsto B\) where \(B= \mathbb{Z}/ p \mathbb{Z}\). Let \(G(k)\) be the subgroup of \(k\)-th powers of elements of \(B\). Depending on the size of \(G(k)\) the authors give an upper bound for \(g\). When \(\vert G(k)\vert \geq 2\) the upper bound equals \(\lceil (1+\frac{1}{p-1})k \rceil\). Under some technical conditions on \(k\) the upper bound is sharpened.