We perform a numerical study of two-dimensional amorphous photonic structures consisting of dielectric rods in a close-packed honeycomb-like structure, with nearest-neighbor rods joined by dielectric veins of various widths. For vein widths comparable to the rod radius, the in-plane Q factors for the transverse magnetic (TM) modes are enhanced by around two orders of magnitude, due to the presence of short-range order in the amorphous lattice, even though the density of states has a weak dip rather than a prominent gap. Such structures may be useful for realizing multi-mode random quantum cascade lasers, which lase in the TM polarization.