Let $$\mathfrak{g }=\mathfrak{s }\mathfrak{l }(1|n+1)$$ be the classical Lie superalgebra of type $$A(0,n)$$ over an algebraically closed field of prime characteristic $$p>2$$ . A sufficient condition is provided for baby Kac $$\mathfrak{g }$$ -modules to be simple. Moreover, simple $$\mathfrak{g }$$ -modules with (quasi) regular semisimple characters are classified. In particular, up to isomorphism, all the simple modules for $$\mathfrak{s }\mathfrak{l }(1|2)$$ are determined, and representatives and dimensions of simples are precisely given. As an application, simple modules for the general linear Lie superalgebra $$\mathfrak{g }\mathfrak{l }(1|n+1)$$ with certain $$p$$ -characters are classified. In particular, a complete classification of simple $$\mathfrak{g }\mathfrak{l }(1|2)$$ -modules is given.