= Solution
The background remains matter dominated, so $H=2/(3t)$ and $4\pi G\bar\rho_m=2/(3t^2)$. For the <reduced gravitational clustering strength>, substitute $\Delta_m\propto t^s$ into the modified equation:
$$
s(s-1)+\frac43s-\frac23(1-f)=0.
$$
The growing root is the <matter-era growth with reduced gravitational clustering strength>
$$
\boxed{s(f)=\frac{-1+\sqrt{25-24f}}6}.
$$
It satisfies $0<s(f)<2/3$ for $0<f<1$. Between $t_f$ and the onset $t_\Lambda$ of dark-energy domination, standard growth supplies $(t_\Lambda/t_f)^{2/3}$ whereas the modified model supplies $(t_\Lambda/t_f)^{s(f)}$. Subsequent growth freezes in both, so
$$
\boxed{
\frac{\Delta_{m,{\rm new}}(t_0)}
{\Delta_{m,{\rm standard}}(t_0)}
\simeq
\left(\frac{t_\Lambda}{t_f}\right)^{s(f)-2/3}}.
$$
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