The phrase "significant influence" refers to having a major impact or control over something or someone. It means that your actions, decisions, or opinions can greatly affect the outcome or direction of a situation. Full definition
They have a genuine and trusted relationship with their followers, resulting in significant influence on consumer decision making. (shoppinglinks.com)
Imagine that... The most significant influence of the second richest sexy old man. (besmartrich.com)
A cousin in a shorter marriage, with considerable legally documented abuse and no kids, but without significant influence in the church, was rejected. (religion.blogs.cnn.com)