The phrase "bear on the effort" is correct and usable in written English.
You can use this phrase to mean that "something affects or influences the effort" or that something "has significant importance related to the effort". For example, "The outcome of the project will bear heavily on the effort of the team."
"What we're doing here at the G20 in part is seeking to gain additional contributions from some of our coalition partners, so that we're able to bring more force to bear on that effort".
None of this will be easy, but Europe brings long, and often hard, experience to bear on these efforts.
The findings bear on natural resource management: If other species prove to be inherently unstable, it could complicate efforts to manage populations.
Though their contribution has been disputed, it's generally thought that the Huguenots brought French expertise to bear on the region's wine-growing efforts and planted many vineyards.
Religious and indigenous leaders from all corners of the globe launched today an unprecedented initiative they say will bring needed moral attention and spiritual commitment to bear on global efforts to end deforestation and protect the tropical rainforests forests that are fundamental to human life, the planet's health and reducing the emissions fueling climate change.
That move is consistent with a larger, bipartisan effort to bring more evidence to bear on the design and implementation of federal programs.
Attorneys general for other states could join in Mr. Schneiderman's efforts, bringing far greater investigative and legal resources to bear on the issue.
Kavanaugh made no apparent effort to bring a lifetime of professional expertise and perspective to bear on the difficult issues under consideration.
We have seen band-aid approaches, but not a serious effort that brings the vast federal resources that could be brought to bear on the problem into action.
