In Correlation Clustering, the input is a graph $G=(V,E)$ with weight function $ω: {V \choose 2}\to Z$ and the task is to partition the vertex set into clusters such that the total weight of edges between clusters and missing edges inside clusters is minimized. Due to close connections between Correlation Clustering and Edge Multicut, deciding whether there is a partition with total cost at most $k$ is FPT with respect to $k$ but a polynomial kernel is presumably impossible. We study the influence of the structure of the fuzzy edge graph, that is, the graph induced by the weight-0 edges, on the problem complexity. We show in particular that Correlation Clustering admits a polynomial problem kernel when parameterized by $k+d$, where $d$ is the degeneracy of the fuzzy edge graph, and when parameterized by $k+c$, where $c$ is the closure of the fuzzy edge graph. We complement these positive results by showing hardness for several settings where the graph induced by the edges and nonedges has very restricted structure.