  Let $p>5$ be a prime. We prove congruences modulo $p^{3-d}$ for sums of the
general form $\sum_{k=0}^{(p-3)/2}\binom{2k}{k}t^k/(2k+1)^{d+1}$ and
$\sum_{k=1}^{(p-1)/2}\binom{2k}{k}t^k/k^d$ with $d=0,1$. We also consider the
special case $t=(-1)^{d}/16$ of the former sum, where the congruences hold
modulo $p^{5-d}$.
