  In this paper, we investigate the non-equilibrium diffusion limit of the
compressible Navier-Stokes-Fourier-P1 (NSF-P1) approximation model at low Mach
number, which arises in radiation hydrodynamics, with general initial data and
a parameter $\delta \in [0,2]$ describing the intensity of scatting effect. In
previous literature, only $\delta =2$ and well-prepared initial data case to
the NSF-P1 model was considered. Here we prove that, for partial general
initial data and $\delta =2$, this model converges to the system of low Mach
number heat-conducting viscous flows coupled with a diffusion equation as the
parameter $\epsilon \rightarrow 0$. Compared to the classical NSF system, the
NSF-P1 model has additional new singular structures caused by the radiation
pressure. To handle these structures, we construct an equivalent pressure and
an equivalent velocity to balance the order of singularity and establish the
uniform estimates of solutions by designating appropriate weighted norms and
carrying out delicate energy analysis. We then obtain the convergence of the
pressure and velocity from the local energy decay of the equivalent pressure
and equivalent velocity. We also briefly discuss the variations of the limit
equations as the scattering intensity changes, i.e., $\delta \in (0,2)$. We
find that, with the weakening of scattering intensity, the ``diffusion
property" of radiation intensity gradually weakens. Furthermore, when the
scattering effect is sufficiently weak ($\delta =0$), we can obtain the
singular limits of the NSF-P1 model with general initial data. To our best
knowledge, this is the first result on the influence of scattering intensity in
the non-equilibrium diffusion limit of the NSF-P1 model.
