I hope what I share would be relevant to your question. Introducing a cut-off in the entanglement entropy $($EE$)$ is a way to regulate the counting of degrees of freedom. First, recall that for any co-dimension 2 space-like surface $B$ in a bulk spacetime, there's a Bousso bound $($with quantum corrections $S_{ent}(B)$$)$ on the integral of the entropy flux "$s$" on the light-sheet $L(B)$ as [1]:
$$\int_{L(B)}s \le \frac{Area(B)}{4G_{N}} + S_{ent}(B) $$
Second, working in the framework of AdS/CFT, we assume the role of (H)EE as a measure of the degrees of freedom by the use of holographic $c$-theorem [2]. Now in general, the #d.o.f of the boundary theory is infinite, so is the area of the asymptotically AdS space. Therefore one can speak of some regulations in order to compare these infinities, by introducing a UV cut-off in the field theory which is equivalent to replacing the continuum by the discrete lattice of cells of the cut-off size. As discussed in [3], one can further assume that each of these cells contain one d.o.f, and also each d.o.f is capable of storing one bit of information. Hence in the calculation of holographic EE, the regulation in $Area(B)$ means introducing an IR cut-off $(z=\epsilon \to 0$ in Poincare or $\rho=\rho_{0} \to \infty$ in global coordinates$)$ in the area functional
$$ Area=\int d^{d-1}\sigma \, \sqrt{det(g_{ij})}$$
Long story short, the cut-off is equivalent to the one bit information $($d.o.f$)$ per Planck area bound. I hope it helps you.
$[1]$ A. Strominger and D. Thompson, "Quantum Bousso bound", Phys. Rev. D 70, 044007 (2004), arXiv:hep-th/0303067.
$[2]$ R. C. Myers and A. Sinha, "Holographic c-theorems in arbitrary dimensions", J. High Energ. Phys. 2011, 125 (2011), arXiv:1011.5819.
$[3]$ L. Susskind and E. Witten, "The Holographic Bound in Anti-de Sitter Space", arXiv:hep-th/9805114.
