Congruences involving the reciprocals of central binomial coefficients

We present several congruences modulo a power of prime $p$ concerning sums of the following type ${\sum }_{k=1}^{p-1}\frac{m^k}{k^r}{\left(\genfrac{}{}{0px}{}{2k}{k}\right)}^{-1}$ which reveal some interesting connections with the analogous infinite series.