"expands its influence" is a correct and usable phrase in written English.
You can use it when you are talking about how an organization, idea, or movement is spreading or increasing its reach. For example, "The company has been expanding its influence, increasing its customer base in the western states."
Adding friends in Southeast Asia expands its influence and fine tunes its balancing act.
As Iran systematically expands its influence in southern Iraq, the Americans are being relegated to a secondary role.
Indeed, as the Latino population expands its influence in Chicago, as in other American cities, visitors won't have to go looking for the Latin beat.
As wireless power expands its influence, the Internet of Things will change the way we operate and carry on our daily lives.
Under Page, Google has not been afraid to be unpopular with its peers if it means expanding its influence with its users.
Its culture has nationalistic elements, and, partly through the vigorous pursuit of economic relationships, it is expanding its influence around the world.
Solution to the Kurdish conflict in Turkey is essential if it wants to expand its influence in Iraq and important if it is to do the same in Syria.
ISIS did not sign the statement, and has faced growing hostility as it seeks to expand its influence.
It clearly wants to expand its influence from the Siberian borders all the way down to the sea-lanes running along Vietnam and the Philippines.
After the organization's creation in Afghanistan, Mr. bin Laden recognized that global unrest presented it with opportunities to expand its influence, as well as pump its financial machine.
The diocese has huge existing commitments and is in financial trouble, but it also wants to expand its influence in the suburbs.O'Connor's unique personality slowed the decline in influence of New York's Catholics.
