We study the effects of energy transport in the Sun by asymmetric dark matter with momentum and velocity-dependent interactions, with an eye to solving the decade-old Solar Abundance Problem. We study effective theories where the dark matter-nucleon scattering cross-section goes as $v_{rm rel}^{2n}$ and $q^{2n}$ with $n = -1, 0, 1 $ or $2$, where $v_{rm rel}$ is the dark matter-nucleon relative velocity and $q$ is the momentum exchanged in the collision. Such cross-sections can arise generically as leading terms from the most basic nonstandard DM-quark operators. We employ a high-precision solar simulation code to study the impact on solar neutrino rates, the sound speed profile, convective zone depth, surface helium abundance and small frequency separations. We find that the majority of models that improve agreement with the observed sound speed profile and depth of the convection zone also reduce neutrino fluxes beyond the level that can be reasonably accommodated by measurement and theory errors. However, a few specific points in parameter space yield a significant overall improvement. A 3-5 GeV DM particle with $sigma_{SI} propto q^2$ is particularly appealing, yielding more than a $6sigma$ improvement with respect to standard solar models, while being allowed by direct detection and collider limits. We provide full analytical capture expressions for $q$- and $v_{rm rel}$-dependent scattering, as well as complete likelihood tables for all models.
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