= Solution
Use $c=1$ and a spatially flat background dominated by the stated perfect fluid, so $4\pi Ga^2\rho_0=3\mathcal H^2/2$. In <Newtonian gauge>, the scalar <energy> and momentum constraints from the linearized <Einstein field equations> combine to eliminate the potential's time derivative and give the stated relativistic Poisson constraint. Its source contains the velocity correction to the coordinate density perturbation. On sub-horizon scales that correction is small for the ordinary growing modes, recovering the <Poisson equation for Newtonian gravity>; near or outside the horizon the gravitational field cannot be sourced by $\delta$ alone. Equality of the two metric potentials also assumes negligible <scalar anisotropic stress>.
The fluid equations are the linear energy-continuity and momentum-Euler parts of $\nabla_\mu T^{\mu\nu}=0$. Here $c_s^2=\delta p/\delta\rho$ is the relevant dimensionless sound-speed squared in the adopted fluid closure; for a constant-$w$ adiabatic barotropic fluid it equals $w$. A fluid with entropy perturbations needs additional <pressure> information. The equations actually use $\theta=\nabla\cdot\mathbf v$ in Fourier space, the <peculiar-velocity divergence>. Calling it simply momentum density is imprecise: it is the divergence of the velocity, with momentum weighted by $\rho_0+p_0$ before normalization. In particular $\theta$ must have inverse-conformal-time units for the displayed continuity equation to be dimensionally consistent.
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