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The interaction between platform motion and atmospheric turbulence plays a critical role in shaping the wake development of floating offshore wind turbines. Large-eddy simulations are performed to investigate the influence of incoming atmospheric boundary layer turbulence intensity on the wake of a floating offshore wind turbine undergoing surge motion. Three inflow turbulence levels and four surge motion frequencies are considered. Increasing inflow turbulence accelerates wake recovery for the fixed turbine through enhanced turbulent momentum transport. For the surge cases, an intermediate surge frequency consistently yields the most effective wake recovery across all inflow turbulence levels. For the high-frequency surge motion, despite possessing the largest motion energy, becomes progressively less effective as turbulence intensity increases, suggesting that strong ambient turbulence constrains the range of surge frequencies that can improve wake recovery. Spectral proper orthogonal decomposition reveals that low-frequency surge motions generate large-scale coherent structures that persist downstream, while structures generated by high-frequency surge motions decay rapidly. Overall, surge frequency, rather than motion energy alone, governs far-wake recovery under turbulent inflow conditions by setting the scale of the generated coherent structures, with high-frequency surge motions producing smaller structures that are more susceptible to disruption by strong ambient turbulence. These findings provide new physical insight into the coupled effects of platform motion and atmospheric turbulence and have important implications for wake modelling of floating offshore wind turbines in realistic environments.