Congruences for central binomial sums and finite polylogarithms

We prove congruences, modulo a power of a prime $p$, for certain finite sums involving central binomial coefficients $\binom{2k}{k}$, partly motivated by analogies with the well-known power series for $(\arcsin z)^2$ and $(\arcsin z)^4$. The right-hand sides of those congruences involve values of the finite polylogarithms $\pounds_d(x)=\sum_{k=1}^{p-1} x^k/k^d$. Exploiting the available functional equations for the latter we compute those values, modulo the required powers of $p$, in terms of familiar quantities such as Fermat quotients and Bernoulli numbers.