  We use 4d ambitwistor string theory to derive new worldsheet formulae for
tree-level conformal supergravity amplitudes supported on refined scattering
equations. Unlike the worldsheet formulae for super-Yang-Mills or supergravity,
the scattering equations for conformal supergravity are not in general refined
by MHV degree. Nevertheless, we obtain a concise worldsheet formula for any
number of scalars and gravitons which we lift to a manifestly supersymmetric
formula using four types of vertex operators. The theory also contains states
with non-plane wave boundary conditions and we show that the corresponding
amplitudes can be obtained from plane-wave amplitudes by applying momentum
derivatives. Such derivatives are subtle to define since the formulae are
intrinsically four-dimensional and on-shell, so we develop a method for
computing momentum derivatives of spinor variables.
