Observe that for some $c \in \mathbb{N}$ and finite alphabet $\Sigma$, $\Sigma^c$ with $|\Sigma^c| = |\Sigma|^c$ is finite.
Therefore
$$L_c = \bigcup_{u \in \Sigma^c} u\Sigma^*u$$
is the union of a finite number of regular languages.
So $L_c$ must be regular/context-free.
Assume that $\Sigma$ and $c$ are chosen as above and that $\sim_c$ is the indistinguishability relation for $L_c$.
By the Myhill-Nerode theorem, $L_c$ is regular if and only if $\Sigma^*/\sim_c$ is finite.
We now prove $L_c$ regular by showing that each string in $\Sigma^*$ is indistinguishable to one of length $\leq 2c$, so the number of classes in $\Sigma^*/\sim_c$ must be bounded by $|\Sigma|^{2c}$.
Choose $w \in \Sigma^*$, if $|w| \leq c$ then $w \sim_c w$.
If $|w| > c$ then there must be $u \in \Sigma^c$, $y \in \Sigma^+$ such that $w = uy$.
Now take the greatest prefix $u_1$ of $u$ such that $y = xu_1$ for some $x \in \Sigma^*$.
If $z \in \Sigma^*$ with $|z| \geq c$, then
$$wz = uyz \in L_c \iff u \text{ is a suffix of } z \iff uu_1z \in L_c.$$
Because $u_1$ is the greatest prefix of $u$ with $w = uxu_1$, it follows that if $|z| < c$ then also
$$wz = uyz \in L_c \iff u_2z = u \text{ for a suffix } u_2 \text{ of } u_1 \iff uu_1z \in L.$$
So $w \sim_c uu_1$ and $|uu_1| \leq |uu| = 2c$, done.
