The smallest parts function associated with ω(q)

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