Primitive monodromy groups of rational functions with one multiple pole

We classify primitive monodromy groups of rational functions of the form $P/Q$, where $Q$ is a polynomial with no multiple roots and $\operatorname{deg}P>\operatorname{deg}Q+1$. There are 17 families of such functions which are not Belyi functions. Only one family from the list contains functions that have five critical values. All the remaining families consist of functions with at most four critical values and constitute one-dimensional strata in the Hurwitz space. We compute the action of the braid group on generators of their monodromy groups and draw the corresponding megamaps.
The result extends the classification of primitive edge rotation groups of weighted trees obtained by the author and Zvonkin and is also a generalization of the classification of primitive monodromy groups of polynomials obtained by P. Müller.