At large energies, the relevant operator does not affect the field theory, so the UV region of the dual geometry of the perturbed CFT (i.e. close to the boundary) is simply AdS. However, when following the RG flow, at some scale the effects of the perturbation set in, this is where the geometry is deformed.
For a mass perturbed CFT below the mass scale, the mass gap prevents pair creation and the field theory "freezes out". This can be (roughly) modelled via AdS/CFT by capping of the holographic coordinate (and thus the spacetime) and restrict it to some range $0<z<z_0$, as everything that happens beyond $z_0$ cannot affect the mass deformed theory.
The entanglement entropy is of course altered by the deformation as pair produced particles are ultimately responsible for this entropy. The Ryu-Takayanagi formula
can capture this reduction of entanglement by cutting off the bulk homologous surfaces (ending on the entangling region on on the boundary) and thus reducing the surface area.
Strictly speaking, capping off is not well defined. There are however related (and well-defined) concepts like soft-wall models.