  In this letter we continue the development of $W$-representations. We propose
several generalizations of the known models, such as the hypergeometric Hurwitz
$\tau$-functions. We construct $W$-representations for multi-character
expansions, which involve a generic number of sets of time variables. We
propose integral representations for such kind of partition functions which are
given by tensor models and multi-matrix models with multi-trace couplings. We
further propose the $\beta$-deformation of the discussed $W$-representation for
the Hurwitz case for two sets of times as well as for the multi-character case.
