  Atkin and Garvan introduced the functions $N_k(n)$ and $M_k(n)$, which denote
the $k$-th moments of ranks and cranks in the theory of partitions. Let
$e_{2r}(n)$ be the $n$-th Fourier coefficient of $E_{2r}(\tau)/\eta(\tau)$,
where $E_{2r}(\tau)$ is the classical Eisenstein series of weight $2r$ and
$\eta(\tau)$ is the Dedekind eta function. Via the theory of quasi-modular
forms, we find that for $k \leq 5$, $N_k(n)$ and $M_k(n)$ can be expressed
using $e_{2r}(n)$ ($2\leq r \leq k$), $p(n)$ and $N_2(n)$. For $k>5$,
additional functions are required for such expressions. For $r\in \{2, 3, 4, 5,
7\}$, by studying the action of Hecke operators on $E_{2r}(\tau)/\eta(\tau)$,
we provide explicit congruences modulo arbitrary powers of primes for
$e_{2r}(n)$. Moreover, for $\ell \in \{5, 7, 11, 13\}$ and any $k\geq 1$, we
present uniform methods for finding nice representations for $\sum_{n=0}^\infty
e_{2r}\left(\frac{\ell^{k}n+1}{24}\right)q^n$, which work for every $r\geq 2$.
These representations allow us to prove congruences modulo powers of $\ell$,
and we have done so for $e_4(n)$ and $e_6(n)$ as examples. Based on the
congruences satisfied by $e_{2r}(n)$, we establish congruences modulo arbitrary
powers of $\ell$ for the moments and symmetrized moments of ranks and cranks as
well as higher order $\mathrm{spt}$-functions.
