Despite being good candidates for the reduction of pollutant emissions from gas turbines, burners operating in Lean Pre-mixed Prevaporized regimes often face stability issues and can be sensitive to perturbations. The swirling flow used to aerodynamically stabilize the flame can also lead to the appearance of a large-scale coherent flow structure known as the Precessing Vortex Core (PVC). In the present study, a swirl-stabilized combustor fed with liquid dodecane is studied at a globally lean operating condition with the help of high-speed diagnostics and Dynamic Mode Decomposition (DMD) as the main post-processing method. It is shown that the trace of a PVC originating inside the injector is still present in the fuel spray at the entrance of the chamber even though the aerodynamical structure itself is not detectable anymore. The perturbation of the fuel spray is then transmitted to the flame through local equivalence ratio fluctuations. It is observed that the PVC trace on the spray and thus on the flame can be suppressed by air flow modulations generated by a siren device. The suppression of this trace is shown to come from a decay of the aerodynamical structure itself rather than by a change in fuel mixing or vaporization. Analysis of the characteristic frequency of the PVC shows a frequency spread indicating a loss of coherence of the structure with the high amplitude air flow rate fluctuations.