  We show that $b$-Hurwitz numbers with a rational weight are obtained by
taking an explicit limit of a Whittaker vector for the $\mathcal{W}$-algebra of
type $A$. Our result is a vast generalization of several previous results that
treated the monotone case, and the cases of quadratic and cubic polynomial
weights. It also provides an interpretation of the associated Whittaker vector
in terms of generalized branched coverings that might be of independent
interest. Our result is new even in the special case $b=0$ that corresponds to
classical hypergeometric Hurwitz numbers, and implies that they are governed by
the topological recursion of Eynard-Orantin. This gives an independent proof of
the recent result of Bychkov-Dunin-Barkowski-Kazarian-Shadrin.
