  We prove the existence of unique weak solutions to an extension of a
Cahn--Hilliard model proposed recently by C.~Liu and H.~Wu (2019), in which the
new dynamic boundary condition is further generalised with an affine linear
relation between the surface and bulk order parameters. As a first approach to
tackle more general and nonlinear relations, we investigate the existence of
unique weak solutions to a regularisation by a Robin boundary condition.
Included in our analysis is the case where there is no diffusion for the
surface order parameter, which causes new difficulties for the analysis of the
Robin system. Furthermore, for the case of affine linear relations, we show the
weak convergence of solutions as the regularisation parameter tends to zero,
and derive an error estimate between the two models. This is supported by
numerical experiments which also demonstrate some non-trivial dynamics for the
extended Liu--Wu model that is not present in the original model.
