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
$1\leqslant {x_{i}\ll }_{\unicode[STIX]{x1D700},k}p^{1/k+\unicode[STIX]{x1D700}}$
.