Wave-influenced deltas are the most abundant delta type and are also potentially the most at-risk to human-caused changes, owing to the effects of wave-driven sediment transport processes and the short timescales on which they operate. Despite this, the processes controlling wave-influenced growth are poorly understood, and the role of fine-grained cohesive sediment (mud) is typically neglected. Here we simulate idealized river deltas in Delft3D across a range of conditions to interrogate how relative wave-influence and fluvial sediment composition impact delta evolution on decadal-millennial timescales. Our simulations capture the barrier-spit formation and accretion process characteristic of prograding wave-influenced deltas, such as those of the Red (Vietnam), Sinu (Colombia), and Coco (Nicaragua) rivers. Barrier-spit accretion exhibits multi-decadal cyclicity driven by subaqueous accumulation of fluvial sediment near river mouths. Using a range of metrics, we quantify how waves and mud influence delta morphology and dynamics. Results show that waves stabilize and simplify channel networks, smooth shorelines, increase shoreline reworking rates, reduce mud retention in the delta plain, and rework mouth bar sediments to form barrier-spits. Higher fluvial mud concentrations produce simpler and more stable distributary networks, rougher shorelines, and limit back-barrier lagoon preservation without altering shoreline reworking rates. Our findings reveal distinct controls on shoreline change between river-dominated and wave-influenced deltas and demonstrate that mud plays a critical role in delta evolution, even under strong wave influence. These insights could enhance paleoenvironmental reconstructions and inform predictions of delta responses to climate and land-use changes.