  The main aim of the paper is to provide analogues of Simpson's correspondence
on singular projective varieties defined over an algebraically closed field of
characteristic $p>0$. There are two main cases.
  In the first case, we consider analogues of numerically flat vector bundles
on a big open subset of a normal projective variety (with arbitrary
singularities). Here we introduce the S-fundamental group scheme for
quasi-projective varieties that admit compactifications with complement of
codimension $\ge 2$. We prove that this group scheme coincides with the
S-funda\-men\-tal group scheme of the regular locus of any of its (small)
projective compactification. In particular, it provides a new invariant for
normal projective varieties isomorphic in codimension $1$.
  In the second case, we consider vector bundles with an integrable
$\lambda$-connection on a normal projective variety $X$ that is (almost)
liftable modulo $p^2$. In this case we restrict to varieties with $F$-liftable
singularities that are analogous to log canonical singularities. We prove that
a semistable Higgs vector bundle on the regular locus of $X$, with
appropriately defined vanishing Chern classes, admits a canonical Higgs--de
Rham flow. This provides an analogue of some results due to D. Greb, S.
Kebekus, B. Taji and T. Peternell. Finally, we give some applications of the
obtained results, also to varieties defined in characteristic zero.
