We present a combined theoretical and experimental study of the still mysterious pseudogap phase in the lightly hole-doped high-T_c superconductor Na_xCa_2-xCuO₂Cl₂ (Na-CCOC), which has a simple I4/mmm crystal structure stable at all doping levels and temperatures. The CuO₂ layers are nearly independent from each other, suggesting a 2D square lattice for which we consider a single-band Hubbard model solved within Cluster Dynamical Mean-Field Theory (C-DMFT) with a continuous-time quantum Monte Carlo solver. The calculations are compared to ARPES data carried out on the undoped parent compound and on lightly doped Na-CCOC samples. We find good agreement of the spectral functions, which show characteristic features of the pseudogap phase such as a well-defined quasi-particle in the nodal point (π/2,π/2) and the opening of a pseudogap in the anti-nodal region (π,0). Moreover, we observe a magnon-like excitation in the C-DMFT data, which we analyze in the light of fluctuation diagnostics and recent RIXS measurements.