The phrase "negligible influence" means that something has very little or almost no impact or effect. It is not significant enough to make a noticeable difference. Full definition
As such, the only real winner was the Liberal Democrats, who went from «pushed into the sea» to «doing well in some areas» and the only real loser was UKIP, who went from «the next big thing» to being of negligible influence. (politics.stackexchange.com)
Urban heat island effects are real but local, and have a negligible influence (less than 0.006 °C per decade over land and zero over the oceans) on these values. (globalwarmingsolved.com)