Summary: 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 complex 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 \(\mathcal{H}^k(\underline{\Omega}_Z^{d-k})\ne 0\). All these results have been known for hypersurfaces in smooth varieties.