The phrase "great influence" means having a strong effect or power on something or someone. It signifies the ability to shape or change outcomes or decisions in a significant way. Full definition
CO2 is good for the environment, and can't even come close to having as great an influence in regulating the planet's energy stores as plain old water. (wattsupwiththat.com)
Any other crypto that has come up after bitcoin has had a company or a founder behind it and the people in charge exert greater influence over their altcoins than anybody else. (crypto-lines.com)
Other research has found that compared with lower - achieving schools, higher - achieving schools provide all stakeholders — including teachers — with greater influence on decisions. (edutopia.org)