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We analyze the mock modular behavior of $\bar{P}_\omega(q)$, a partitionfunction introduced by Andrews, Dixit, Schultz, and Yee. This function arose ina study of smallest parts functions related to classical third order mock thetafunctions, one of which is $\omega(q)$. We find that the modular completion of$\bar{P}_\omega(q)$ is not simply a harmonic Maass form, but is instead thederivative of a linear combination of products of various harmonic Maass formsand theta functions. We precisely describe its behavior under modulartransformations and find that the image under the Maass lowering operator liesin a relatively simpler space.

On a modularity conjecture of Andrews, Dixit, Schultz, and Yee for a variation of Ramanujan's $\omega(q)$