  We construct renormalized holographic entanglement entropy (HEE) and
subregion complexity (HSC) in the CV conjecture for asymptotically AdS$_4$ and
AdS$_5$ geometries under relevant perturbations. Using the holographic
renormalization method developed in the gauge/gravity duality, we obtain
counter terms which are invariant under coordinate choices. We explicitly
define different forms of renormalized HEE and HSC, according to conformal
dimensions of relevant operators in the $d=3$ and $d=4$ dual field theories. We
use a general embedding for arbitrary entangling subregions and showed that any
choice of the coordinate system gives the same form of the counter terms, since
they are written in terms of curvature invariants and scalar fields on the
boundaries. We show an explicit example of our general procedure. Intriguingly,
we find that a divergent term of the HSC in the asymptotically AdS$_5$ geometry
under relevant perturbations with operators of conformal dimensions in the
range $0< \Delta < \frac{1}{2}\,\, {\rm and} \,\, \frac{7}{2}< \Delta < 4$
cannot be cancelled out by adding any coordinate invariant counter term. This
implies that the HSCs in these ranges of the conformal dimensions are not
renormalizable covariantly.
