To manipulate a situation, especially by asking favours of others; to use one's influence with others to attain a desired goal.
He has the job not because of talent, but because his dad pulled strings with the boss.
(often with “the” or a possessive adjective (such as “his”) before “strings”, pull one's strings) To control a person, organization, or situation by operating behind the scenes, as a puppeteer controls a marionette.