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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 $δ\in [0,2]$ describing the intensity of scatting effect. In previous literature, only $δ=2$ and well-prepared initial data case to the NSF-P1 model was considered. Here we prove that, for partial general initial data and $δ=2$, this model converges to the system of low Mach number heat-conducting viscous flows coupled with a diffusion equation as the parameter $ε\rightarrow 0$. 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., $δ\in (0,2)$. 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 ($δ=0$), 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.