  We study Hardy inequalities for $p$-Schr\"odinger operators on general
weighted graphs. Specifically, we prove a Maz'ya-type result, where we
characterize the space of Hardy weights for $ p $-Schr\"odinger operators via a
generalized capacity. The novel ingredient in the proof is the demonstration
that the simplified energy of the $ p $-Schr\"odinger energy functional is
compatible with certain normal contractions. As a consequence, we obtain a
necessary integrability criterion for Hardy weights. Finally, using some tools
of criticality theory, we investigate the existence of minimizers in the Hardy
inequalities and discuss relations to Cheeger type estimates.
