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Let $\mathfrak{g}=\mathfrak{g}_{\bar0}+\mathfrak{g}_{\bar1}$ be a basicclassical Lie superalgebra over $\mathbb{C}$, and$e=e_{\theta}\in\mathfrak{g}_{\bar0}$ with $-\theta$ being a minimal root of$\mathfrak{g}$. Set $U(\mathfrak{g},e)$ to be the minimal finite$W$-superalgebras associated with the pair $(\mathfrak{g},e)$. In this paper westudy the highest weight theory for $U(\mathfrak{g},e)$, introduce the Vermamodules and give a complete isomorphism classification of finite-dimensionalirreducible modules, via the parameter set consisting of pairs of weights andlevels. Those Verma modules can be further described via parabolic inductionfrom Whittaker modules for $\mathfrak{osp}(1|2)$ or $\mathfrak{sl}(2)$respectively, depending on the detecting parity of$\textsf{r}:=\dim\mathfrak{g}(-1)_{\bar1}$. We then introduce and investigatethe BGG category $\mathcal{O}$ for $U(\mathfrak{g},e)$, establishing highestweight theory, as a counterpart of the works for finite $W$-algebras byBrundan-Goodwin-Kleshchev and Losev, respectively.  In comparison with the non-super case, the significant difference here liesin the situation when $\textsf{r}$ is odd, which is a completely newphenomenon. The difficulty and complicated computation arise from there.

Highest weight theory for minimal finite $W$-superalgebras and related Whittaker categories