Schoenfield's absoluteness states that if $\phi$ is $\Sigma^1_2$ then $V\models \phi$ iff $L\models \phi$. The set of reals in $L$ is $\Sigma^1_2$ and it is the largest countable $\Sigma^1_2$ set of reals if $\omega_1 ^L < \omega_1$.

If $\phi$ is $\Sigma^1_4$ then $V\models \phi$ iff $\mathcal M_2 \models \phi$, where $\mathcal M_2$ is the minimal proper class mouse with $2$ Woodins. The largest countable $\Sigma^1_4$ set of reals is exactly the set of reals in $\mathcal M_2$.

In general the largest countable $\Sigma^1_{2n+1}$ set of reals is exactly the set of reals in the minimal proper class mouse with $n$ Woodins $\mathcal M_n$. Could you redirect me to a reference please, I would like to see a proof.

Also how far can this phenomenon be pushed in general? For example, if $\phi$ is a second order formula (say $\Sigma^2_1$), how many Woodins would we need so that $\phi$ is absolute between $V$ and the appropriate proper class mouse containing these Woodin cardinals? Would the reals of that proper class mouse necessarily be the largest countable $\Sigma^2_1$ set of reals ( if it exists, I don't know if it does)?