"use its leverage" is correct and can be used in written English.
This phrase is usually used to refer to the use of a particular resource or advantage to create a beneficial situation. For example, "The company used its leverage in the new market to make more efficient investments."
He envisioned a proactive federal government that would use its leverage to transform failed programs and ingrained habits.
The group also said it would use its leverage to reshape the health insurance market.
It could use its leverage and tap into such programmes to help fighters gain the local support needed to defeat Isis.
Baylor should drop its potential legal squabble, although it could use its leverage to help the league squeeze a bigger buyout from the Aggies.
It needs to use its leverage — now.
I think the U.S. is unwilling to use its leverage because, frankly, it's not comfortable with the role of foreign occupier.
It needs to use its leverage over the actions of others: other businesses or even, within reason, the actions of governments.
It is eager to use its leverage to reduce the level of violence and bring about a lasting peace in Syria.
The United States has hoped that China would use its leverage over North Korea to restrain it from any further attacks, but so far China has not rebuked the North's leaders, at least in public.
Moscow will likely use its leverage to enhance its own standing in the region". It may have to use that leverage soon in order to stave off a deadly spiral of violence.
The report also claims that McDonald's "failed to use its leverage" with Taylor Farms to address the alleged violations, and that it ultimately "caused further damage to the workers" in the way the fast-food giant intervened.
