Let $t:{\mathbb F_p} \to C$ be a complex valued function on ${\mathbb F_p}$. A classical problem in analytic number theory is bounding the maximum
$$M(t): = \mathop {\max }\limits_{0 \le H < p} \left| {{1 \over {\sqrt p }}\sum\limits_{0 \le n < H} {t(n)} } \right|$$
of the absolute value of the incomplete sums
$(1/\sqrt p )\sum\nolimits_{0 \le n < H} {t(n)} $. In this very general context one of the most important results is the Pólya–Vinogradov bound
$$M(t) \le {\left\| {\hat t} \right\|_\infty }\log 3p,$$
where
$\hat t:{\mathbb F_p} \to \mathbb C$ is the normalized Fourier transform of
t. In this paper we provide a lower bound for certain incomplete Kloosterman sums, namely we prove that for any
$\varepsilon > 0$ there exists a large subset of
$a \in \mathbb F_p^ \times $ such that for
$${\rm{k}}{1_{a,1,p}}:x \mapsto e((ax + \bar x)/p)$$ we have
$$M({\rm{k}}{1_{a,1,p}}) \ge \left( {{{1 - \varepsilon } \over {\sqrt 2 \pi }} + o(1)} \right)\log \log p,$$
as
$p \to \infty $. Finally, we prove a result on the growth of the moments of
${\{ M({\rm{k}}{1_{a,1,p}})\} _{a \in \mathbb F_p^ \times }}$.