set theory - Definable surjections in $J$-structures

Recall that a $J$-structure is an amenable structure of the form ($J_{alpha}^A,B$) where $A$ and $B$ are predicates and $alpha$ is a limit ordinal. Then if you let $M=J_{alpha}^A$, there is a surjective function $f:[alpha]^{lneq omega} rightarrow M$ which is $Sigma_1^M$. Can we prove that there exists surjective functions $f:[alpha]^{lneq omega} rightarrow M$ which are $Sigma_n^M$ for all $n$<$omega$?

The proof that there is a surjective function $f:[alpha]^{lneq omega} rightarrow M$ which is $Sigma_1^M$ uses that $h_M(alpha)$ is a $Sigma_1$ elementary substructure of $M$ ($h_M(alpha)$ the Skolem hull of $alpha$)