Graph minors XV. Giant steps

Let G be a graph with a subgraph H drawn with high representativity on a surface Σ. When can the drawing of H be extended "up to 3-separations" to a drawing of G in Σ if we permit a bounded number (κ say) of "vortices" in the drawing of G, that is, local areas of non-planarity? (The case κ = 0 was studied in the previous paper of this series.) For instance, if there is a path in G with ends in H, far apart, and otherwise disjoint from H, then no such extension exists. We are concerned with the converse; if no extension exists, what can we infer about G? It turns out that either there is a path as above, or one of two other obstructions is present.