We prove that for any positive integers $k,n$ with $n>\frac{3}{2}(k^{2}+k+2)$, prime $p$, and integers $c,a_{i}$, with $p\nmid a_{i}$, $1\leqslant i\leqslant n$, there exists a solution $\text{}\underline{x}$ to the congruence $$\begin{eqnarray}\mathop{\sum }_{i=1}^{n}a_{i}x_{i}^{k}\equiv c\hspace{0.6em}({\rm mod}\hspace{0.2em}p)\end{eqnarray}$$ with $1\leqslant {x_{i}\ll }_{k}p^{1/k}$, $1\leqslant i\leqslant n$. This upper bound is best possible. Refinements are given for smaller $n$, and for variables restricted to intervals in more general position. In particular, for any $\unicode[STIX]{x1D700}>0$ we give an explicit constant $c_{\unicode[STIX]{x1D700}}$ such that if $n>c_{\unicode[STIX]{x1D700}}k$, then there is a solution with $1\leqslant {x_{i}\ll }_{\unicode[STIX]{x1D700},k}p^{1/k+\unicode[STIX]{x1D700}}$.