The main aim of this paper is to investigate approximation and convergence of degenerate $(a,k)$-regularized $C$-resolvent families in locally convex spaces. We continue our previous research of non-degenerate case $[$Numer. Funct. Anal. Appl. {\bf 35} $(2014)$, $1579-1606]$, following the multivalued linear operators approach to abstract degenerate differential equations. We also consider Laguerre expansions of degenerate $(a,k)$-regularized $C$-resolvent families.