  We derive identities from Hecke operators acting on a family of
Eisenstein-eta quotients, yielding congruences for their coefficients modulo
powers of primes. As an application we derive systematic congruences for
several higher-order smallest parts functions modulo prime powers, resolving a
question of Garvan for these cases. We also relate moments of cranks and ranks
to the partition function modulo prime powers. Some of our results strengthen
and generalize those of a 2023 paper by Wang and Yang.
