  In this paper we shall assume that the ambient manifold is a
pseudo-Riemannian space form $N^{m+1}_t(c)$ of dimension $m+1$ and index $t$
($m\geq2$ and $1 \leq t\leq m$). We shall study hypersurfaces $M^{m}_{t'}$
which are polyharmonic of order $r$ (briefly, $r$-harmonic), where $r\geq 3$
and either $t'=t$ or $t'=t-1$. Let $A$ denote the shape operator of
$M^{m}_{t'}$. Under the assumptions that $M^{m}_{t'}$ is CMC and $Tr A^2$ is a
constant, we shall obtain the general condition which determines that
$M^{m}_{t'}$ is $r$-harmonic. As a first application, we shall deduce the
existence of several new families of proper $r$-harmonic hypersurfaces with
diagonalizable shape operator, and we shall also obtain some results in the
direction that our examples are the only possible ones provided that certain
assumptions on the principal curvatures hold. Next, we focus on the study of
isoparametric hypersurfaces whose shape operator is non-diagonalizable and also
in this context we shall prove the existence of some new examples of proper
$r$-harmonic hypersurfaces ($r \geq 3$). Finally, we shall obtain the complete
classification of proper $r$-harmonic isoparametric pseudo-Riemannian surfaces
into a $3$-dimensional Lorentz space form.
