  A complete canonical formulation of general covariance makes it possible to
construct new modified theories of gravity that are not of higher-curvature
form, as shown here in a spherically symmetric setting. The usual uniqueness
theorems are evaded by using a crucial and novel ingredient, allowing for
fundamental fields of gravity distinct from an emergent space-time metric that
provides a geometrical structure to all solutions. As specific examples, there
are new expansion-shear couplings in cosmological models, a form of modified
Newtonian dynamics (MOND) can appear in a space-time covariant theory without
introducing extra fields, and related effects help to make effective models of
canonical quantum gravity fully consistent with general covariance.
