  We reformulate the twistor construction for hyper- and quaternion-K\"ahler
manifolds, introducing new sigma models that compute scalar potentials for the
geometry. These sigma models have the twistor space of the quaternionic
manifold as their target and encode finite non-linear perturbations of the flat
structures. In the hyperk\"ahler case our twistor sigma models compute both
Plebanski fundamental forms (including the K\"ahler potential), while in the
quaternion-K\"ahler setting the twistor sigma model computes the K\"ahler
potential for the hyperk\"ahler structure on non-projective twistor space. In
four-dimensions, one of the models provides the generating functional of
tree-level MHV graviton scattering amplitudes; perturbations of the
hyperk\"ahler structure corresponding to positive helicity gravitons. The sigma
model's perturbation theory gives rise to a sum of tree diagrams observed
previously in the literature, and their summation via a matrix tree theorem
gives a first-principles derivation of Hodges' formula for MHV graviton
amplitudes directly from general relativity. We generalise the twistor sigma
model to higher-degree (defined in the first case with a cosmological
constant), giving a new generating principle for the full tree-level graviton
S-matrix.
