  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
$6\sigma$ 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.
