  Cosmology may give rise to appreciable populations of both particle dark
matter and primordial black holes (PBH) with the combined mass density
providing the observationally inferred value $\Omega_{\rm DM}\approx0.26$.
However, previous studies have highlighted that scenarios with both particle
dark matter and PBH are strongly excluded by $\gamma$-ray limits for particle
dark matter with a velocity independent thermal cross section $\langle\sigma
v\rangle\sim3\times10^{-26}{\rm cm}^3/{\rm s}$, as is the case for classic WIMP
dark matter. Here we extend these existing studies on $s$-wave annihilating
particle dark matter to ascertain the limits from diffuse $\gamma$-rays on
velocity dependent annihilations which are $p$-wave with $\langle\sigma v
\rangle\propto v^2$ or $d$-wave with $\langle\sigma v \rangle\propto v^4$,
which we find to be considerably less constraining. Furthermore, we highlight
that even if the freeze-out process is $p$-wave it is relatively common for
(loop/phase-space) suppressed $s$-wave processes to actually provide the
leading contributions to the experimentally constrained $\gamma$-ray flux from
the PBH halo. This work also utilyses a refined treatment of the PBH dark
matter density profile and outlines an improved application of extra-galactic
$\gamma$-ray bounds.
