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The notion of Cauchy dual for left-invertible covariant representations wasstudied in \cite{HS19}. Using the Moore-Penrose inverse, we extend this notionfor the covariant representations having closed range and explore severaluseful properties. Based on \cite{EMZ21}, we obtain a Wold-type decompositionfor {regular} completely bounded covariant representation whose Moore-Penroseinverse is regular. Also, we discuss an example related to the non-commutativebilateral weighted shift considered in \cite{AHS22}. We prove that the Cauchydual of the concave covariant representation $(\sigma, V)$ modulo $N(\wV)$ ishyponormal modulo $N(\wV)$.

Cauchy dual and Wold-type decomposition for bi-regular covariant representations