Classification of certain $n$-Derivations of Hom-Lie superalgebras

Let $\chi$ be a Hom-Lie superalgebra and $n\ge 2$ be a fixed integer. In this paper, we study the Lie $n$-derivation algebra $hN\mathrm{Der}(\chi )$ as well as the generalized Lie $n$-derivation algebra $hGN\mathrm{Der}(\chi )$. We also analyze the structures of the $n$-centroids, the $n$-quasi centroids, and the central $n$-derivations, and examine how they relate to $hN\mathrm{Der}(\chi )$ and $hGN\mathrm{Der}(\chi )$. Moreover, we obtain a full classification of $n$-derivations of Hom-Lie superalgebras and demonstrate that, for simple Hom-Lie superalgebras, one has $hN\mathrm{Der}(\chi )=h\mathrm{Der}(\chi )$.