McMullen has proved that the Julia set of $z^n+\lambda/z^d$ is a Cantor of Jordan curves as soon as the local degrees $n$ and $d$ satisfy a certain arithmetic condition (and $|\lambda|>0$ is small enough). Many other authors have studied similar examples obtained by adding singular perturbations to a polynomial. I will introduce a general definition of the so called McMullen-like mappings that unifies this behavior. Every topological conjugation class is described by the data of a postcritically finite hyperbolic polynomial and a collection of local degrees, each associated with a periodic Fatou domain. Using this invariant, the arithmetic condition for existence can be generalized. I will show that this condition is actually necessary by using the theory of Thurston's obstructions.