  Let $\chi$ be a primitive character modulo $q$, and let $\delta > 0$.
Assuming that $\chi$ has large order $d$, for any $d$th root of unity $\alpha$
we obtain non-trivial upper bounds for the number of $n \leq x$ such that
$\chi(n) = \alpha$, provided $x > q^{\delta}$. This improves upon a previous
result of the first author by removing restrictions on $q$ and $d$. As a
corollary, we deduce that if the largest prime factor of $d$ satisfies $P^+(d)
\to \infty$ then the level set $\chi(n) = \alpha$ has $o(x)$ such solutions
whenever $x > q^{\delta}$, for any fixed $\delta > 0$.
  Our proof relies, among other things, on a refinement of a mean-squared
estimate for short sums of the characters $\chi^\ell$, averaged over $1 \leq
\ell \leq d-1$, due to the first author, which goes beyond Burgess' theorem as
soon as $d$ is sufficiently large. We in fact show the alternative result that
either (a) the partial sum of $\chi$ itself, or (b) the partial sum of
$\chi^\ell$, for ``almost all'' $1 \leq \ell \leq d-1$, exhibits cancellation
on the interval $[1,q^{\delta}]$, for any fixed $\delta > 0$.
  By an analogous method, we also show that the P\'{o}lya-Vinogradov inequality
may be improved for either $\chi$ itself or for almost all $\chi^\ell$, with $1
\leq \ell \leq d-1$. In particular, our averaged estimates are non-trivial
whenever $\chi$ has sufficiently large even order $d$.
