  The notion of Cauchy dual for left-invertible covariant representations was
studied by Trivedi and Veerabathiran. Using the Moore-Penrose inverse, we
extend this notion for the covariant representations having closed range and
explore several useful properties. We obtain a Wold-type decomposition for
{regular} completely bounded covariant representation whose Moore-Penrose
inverse is regular. Also, we discuss an example related to the non-commutative
bilateral weighted shift. We prove that the Cauchy dual of the concave
covariant representation $(\sigma, V)$ modulo $N(\wV)$ is hyponormal modulo
$N(\wV)$.
