What word describes the act of influencing someone through gentle persuasion?

The word that aptly describes the act of influencing someone through gentle persuasion is "wheedle." This term specifically refers to using flattery, coaxing, or soft talk to persuade someone to do something or to gain something, often in a light-hearted or playful manner. It implies a sense of charm and persuasion rather than force, which aligns with the context of gentle influence.

In contrast, the other options do not convey this meaning. "Vitiate" refers to impairing or weakening something, often used in a legal or moral sense. "Fatuous" means silly or foolish, lacking intelligence or awareness, which does not pertain to persuasion at all. "Quiescence" denotes a state of inactivity or dormancy, which does not relate to the act of influencing others. Hence, "wheedle" stands out as the most appropriate choice for gentle persuasion.