  We consider a renormalizable extension of the standard model whose fermionic
dark matter (DM) candidate interacts with a real singlet pseudo-scalar via a
pseudo-scalar Yukawa term while we assume that the full Lagrangian is
CP-conserved in the classical level. When the pseudo-scalar boson develops a
non-zero vacuum expectation value, spontaneous CP-violation occurs and this
provides a CP-violated interaction of the dark sector with the SM particles
through mixing between the Higgs-like boson and the SM-like Higgs boson. This
scenario suggests a minimal number of free parameters. Focusing mainly on the
indirect detection observables, we calculate the dark matter annihilation cross
section and then compute the DM relic density in the range up to $m_{\text{DM}}
= 300$ GeV. We then find viable regions in the parameter space constrained by
the observed DM relic abundance as well as invisible Higgs decay width in the
light of 125 GeV Higgs discovery at the LHC. We find that within the
constrained region of the parameter space, there exists a model with dark
matter mass $m_{\text{DM}} \sim 38$ GeV annihilating predominantly into $b$
quarks, which can explain the Fermi-LAT galactic gamma-ray excess.
