The k-Cycle Perfect Matching Reconfiguration Problem (k-Cycle-PM-RP) concerns the task of deciding if a pair of perfect matchings for a simple graph can be interconverted via a series of local rearrangements that allow for transitions between perfect matchings if and only if the symmetric difference of their edge sets induces a simple cycle of length \(k \in 2\mathbb {N}_{>1}\) (Bonamy et al.; Proc. 44th MFCS; 2019). While it is known that the k-Cycle-PM-RP is PSPACE-complete in the case of bounded bandwidth and bounded degree bipartite graphs, as well as polynomial time solvable in the case of outerplanar graphs, cographs, and strongly orderable graphs, it has remained an open question whether the problem is fixed-parameter tractable in the genus of a graph. In this work, we answer this question by showing that the k-Cycle-PM-RP remains PSPACE-complete for bounded bandwidth and bounded degree planar graphs for every \(k \in 2\mathbb {N}_{>1}\) . On the other hand, among some other positive results, we show that it is fixed-parameter tractable in a parameter \(r \in \mathbb {N}\)  to add at most r edges to a planar graph G to construct a strongly orderable planar graph \(G'\) , where we can observe that the \(\left( k=4\right) \) -Cycle-PM-RP on strongly orderable graphs is both polynomial time tractable and guaranteed to admit a positive solution.