We establish two families of congruences modulo powers of 5 for the Fourier coefficients of $(2E_2(2τ)-E_2(τ))η(2τ)^{-1}$, where $E_2(τ)$ is the weight 2 Eisenstein series and $η(τ)$ is the Dedekind eta function. This allows us to prove similar congruences for two smallest parts functions. The first function is $\mathrm{spt}_ω(n)$, which was introduced by Andrews, Dixit and Yee and associated with Ramanujan/Watson's third order mock theta function $ω(q)$. The second one is $\mathrm{spt}_{C5}(n)$, which appeared in the work of Garvan and Jennings-Shaffer. Moreover, we confirm two conjectural congruences of Wang.