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The competition between Kitaev and Heisenberg interactions away from halffilling is studied for the the hole-doped Kitaev-Heisenberg $t$-$J_K$-$J_H$model on a honeycomb lattice. While the isotropic Heisenberg coupling supportsa time-reversal violating d-wave singlet state, we find that the Kitaevinteraction favors a time-reversal invariant p-wave superconducting phase,which obeys the rotational symmetries of the microscopic model, and is robustfor $J_H<J_K/2$. Within the p-wave superconducting phase, a critical chemicalpotential $\mu_c \approx t$ separates a topologically trivial phase for $|\mu|<\mu_c$ from a topologically non-trivial $Z_2$ time-reversal invariantspin-triplet phase for $|\mu|>\mu_c$.

Competition between d-wave and topological p-wave superconductivity in the doped Kitaev-Heisenberg model