have a influence on sb 还是have a influence with sb

influence on强调一方对另一方单向的影响,前者的地位高于后者。
例如：American influence on the Middle East is past its peak
美国对中东的影响力已经大不如前了
influence with强调双方互相之间的一种关系
As secretary of labor, Frances Perkins used her considerable influence with
Franklin D. Roosevelt to prevent him from restraining strikes by longshoremen and automobile workers.
显然frances perkins这个女的对罗斯福总统的影响力不可能那么强大，他们两人之间有互相影响的关系，是平等的。但是这个句子主要强调的当然是这个女的对罗斯福的影响。用influence with.
又如：
Middle East shifts may weaken Iran' s influence with Palestinians
Poland trims Russian influence with Qatar gas loan deal

have a influence on sb.

1.influence意为“影响”或“感化”时,后面可接upon或on.This example has great influence upon /on young people.这个事例对年轻人有重大的影响.2.区别下面两个句子：Mike used to have great influence over that...

是have a influence on sb.
是对某人有影响。