Why can broad ash deposits around a submarine volcano suggest that some vents reached the sea surface?

Answer: Air lets ash travel farther because it slows falling fragments less than water does.

Currents let ash travel farther because they sweep loose particles across the seabed. — Seafloor currents can sweep loose ash after it settles, so reworking is a real process. But currents follow the seabed and tend to modify the surface; they do not build a broad layered apron around the vent. The wide, layered deposit is better explained by material carried outward from an eruption column.

Water lets ash travel farther because it keeps eruption particles aloft much longer. — Water can keep fine ash suspended for a time, so it seems able to carry ash far. But dense water also drags coarse fragments down and makes eruption-fed currents collapse near the vent. Broad primary aprons spreading far from the vent fit air transport better than water transport.

Air lets ash travel farther because it slows falling fragments less than water does. ✓ — When an eruption breaches the sea surface, ash enters air instead of thick water. Air slows falling fragments far less than water, so ash and debris can spread outward before settling. That is why a broad ash deposit can hint at vents reaching the surface even without a tall cone.