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 2kk, partly motivated by analogies with the well-known power series for ( z)² and ( z)⁴. The right-hand sides of those congruences involve values of the finite polylogarithms d(x)=∑ₖ₌₁ᵖ⁻¹ xᵏ/kᵈ. 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.