The phrase "have influence on" is correct and usable in written English. You can use it when talking about a person or thing that has an effect on a particular situation or outcome. For example, "In this community, parents have influence on what their children do and learn.".
You have influence on her.
Doesn't that have influence on young men?
"If we want to have influence on the island, what makes sense?" he said.
"Now you realise that you don't really have influence on the dynamics of the systems in which you are.
She told the Guardian: "How can we expect to have influence on standards around the world when we do this?
Their hope now is to send a few delegates to the party's convention and have influence on the party platform.
After all, electoral-representative democracy is a set of arrangements that enable ordinary citizens to have influence on government.
Being a terminologist, I care about word choice. Ludwig simply helps me pick the best words for any translation. Five stars!
Maria Pia Montoro
Terminologist and Q/A Analyst @ Translation Centre for the Bodies of the European Union