<p>We construct a one-parameter family of Lorentzian conformal structures on the canonical circle bundle of a partially integrable contact almost Cauchy–Riemann manifold. This builds on previous work by Leitner, who generalised Fefferman’s construction associated to a CR manifold to the non-involutive case. We provide characterisations of these conformal structures and show that they admit distinguished pure spinor fields. We introduce exact ‘perturbations’ of such Fefferman spaces by a semi-basic one-form, which can be suitably interpreted as a tuple of weighted tensors on the almost CR manifold. The resulting perturbed conformal space is an instance of a so-called nearly Robinson manifold introduced recently by Fino, Leistner and the present author. We investigate the existence of metrics in these conformal classes which satisfy appropriate subsystems of the Einstein equations. These metrics are defined only off cross-sections of Fefferman’s circle bundle, and are conveniently expressed in terms of almost Lorentzian densities, which include Gover’s almost Einstein scales as a special case. In particular, in dimensions greater than four, almost Einstein scales always arise from so-called CR–Einstein structures, almost CR analogues of Einstein metrics. We derive necessary and sufficient conditions for a perturbed Fefferman space to be conformally flat on the zero set of an almost Einstein scale. We construct an explicit example of a CR–Einstein structure on a strictly almost CR five-manifold based on a strictly almost Kähler–Einstein four-manifold due to Nurowski and Przanowski.</p>