  We show that if $Z$ is a local complete intersection subvariety of a smooth
complex variety $X$, of pure codimension $r$, then $Z$ has $k$-rational
singularities if and only if $\widetilde{\alpha}(Z)>k+r$, where
$\widetilde{\alpha}(Z)$ is the minimal exponent of $Z$. We also characterize
this condition in terms of the Hodge filtration on the intersection cohomology
Hodge module of $Z$. Furthermore, we show that if $Z$ has $k$-rational
singularities, then the Hodge filtration on the local cohomology sheaf
$\mathcal{H}^r_Z(\mathcal{O}_X)$ is generated at level $\dim(X)-\lceil
\widetilde{\alpha}(Z)\rceil-1$ and, assuming that $k\geq 1$ and $Z$ is
singular, of dimension $d$, that $\mathcal{H}^k(\underline{\Omega}_Z^{d-k})\neq
0$. All these results have been known for hypersurfaces in smooth varieties.
