  The launching of astrophysical jets provides the most compelling
observational evidence for direct extraction of black hole (BH) spin energy via
the Blandford-Znajek (BZ) mechanism. Whilst it is know that spinning Kerr BHs
within general relativity (GR) follow the BZ jet power relation, the nature of
BH energy extraction in general theories of gravity has not been adequately
addressed. This study performs the first comprehensive investigation of the BZ
jet power relation by utilising a generalised BH spacetime geometry which
describes parametric deviations from the Kerr metric of GR, yet recovers the
Kerr metric in the limit that all deviation parameters vanish. Through
performing and analysing an extensive suite of three-dimensional covariant
magnetohydrodynamics (MHD) simulations of magnetised gas accretion onto these
generalised BH spacetimes we find that the BZ jet power relation still holds,
in some instances yielding jet powers far in excess of what can be produced by
even extremal Kerr BHs. It is shown that the variation of the quadrupole moment
of the BH can enhance or suppress the effects of BH spin, and by extension of
frame-dragging. This variation greatly enhances or suppresses the observed jet
power and underlying photon ring image asymmetry, introducing a previously
unexplored yet important degeneracy in BH parameter inference.
