  We study one-dimensional strongly interacting quantum gas mixtures, including
both the Bose-Fermi and spin-1/2 Fermi-Fermi mixtures, with weak p-wave
interactions between intra-component fermions, and demonstrate that the weak
p-wave interaction can not be omitted in the strongly interacting regime where
the strength of p-wave interactions is comparable with the inverse of the
strength of strongly repulsive s-wave interactions. While the total density
distribution is not sensitive to the weak p-wave interaction, we find that the
p-wave interaction plays an important role in determining the species-dependent
(or spin-dependent) density distributions and produces significant physical
effects on the low-energy spin dynamics. We also derive effective spin-exchange
models for strongly interacting quantum gas mixtures with weak p-wave
interactions, which indicate that a quantum phase transition from
anti-ferromagnetic state to ferromagnetic state can be induced by tuning the
relative strengths of intra-component and inter-component interactions.
