  We present a method for obtaining congruences modulo powers of a prime
number~$p$ for combinatorial sequences whose generating function satisfies an
algebraic differential equation. This method generalises the one by Kauers and
the authors [Electron. J. Combin. 8(2) (2012), Art. P37; arXiv:1107.2015] from
$p=2$ to arbitrary primes. Our applications include congruences for numbers of
non-crossing graphs and numbers of Kreweras walks modulo powers of~$3$, as well
as congruences for Fu\ss-Catalan numbers and blossom tree numbers modulo powers
of arbitrary primes.
