  We study the simplicity of Kac induced modules over the $\ell$-th Takiff
superalgebras $\widetilde{\mathfrak g}_\ell:= \widetilde{\mathfrak g}\otimes
\mathbb C[\theta]/(\theta^{\ell+1})$, for $\ell>0$, associated with the Lie
superalgebras $\widetilde{\mathfrak g}$ of type I. We formulate a general
notion of typical weights and typical Jordan blocks of the category $\mathcal
O$ for $\widetilde{\mathfrak g}_\ell$ associated with Lie superalgebras
$\mathfrak{gl}(m|n)$, $\mathfrak{osp}(2|2n)$ and $\mathfrak{pe}(n)$. For Lie
superalgebras $\mathfrak{gl}(m|n)$ and $\mathfrak{osp}(2|2n)$, we establish an
equivalence from an arbitrary typical Jordan block of the category $\mathcal O$
for $\widetilde{\mathfrak g}_\ell$ to a Jordan block of the category $\mathcal
O$ for the even subalgebra of $\widetilde{\mathfrak g}_\ell$. This provides a
solution to the problem of determining the composition multiplicities of the
Verma modules over $\widetilde{\mathfrak g}_\ell$ with typical highest weights.
We also investigate non-singular Whittaker modules over these Takiff
superalgebras. In particular, we obtain a classification of non-singular simple
Whittaker modules and a criterion for simplicity of non-singular standard
Whittaker modules.
