  We consider the sums of the form $$ S=\sum_{x=1}^{N}
\exp\big((ax+b_1g_1^x+... +b_rg_r^x)/p \big) $$, where $p$ is prime and
$g_1,..., g_r$ are primitive roots $\pmod p$. An almost forty years old problem
of L. J. Mordell asks to find a nontrivial estimate of $S$ when at least two of
the coefficients $b_1,...,b_r$ are not divizible by $p$. Here we obtain a
nontrivial bound of the average of these sums when $g_1$ runs over all
primitive roots $\pmod p$.
