  We investigate the possibility of improving the $p$-Poincar\'e inequality
$\|\nabla_{\mathbb{H}^N} u\|_p \ge \Lambda_p \|u\|_p$ on the hyperbolic space,
where $p>2$ and $\Lambda_p:=[(N-1)/p]^{p}$ is the best constant for which such
inequality holds. We prove several different, and independent, improved
inequalities, one of which is a Poincar\'e-Hardy inequality, namely an
improvement of the best $p$-Poincar\'e inequality in terms of the Hardy weight
$r^{-p}$, $r$ being geodesic distance from a given pole. Certain
Hardy-Maz'ya-type inequalities in the Euclidean half-space are also obtained.
