The phrase "a core responsibility" is correct and usable in written English.
You can use it to refer to a fundamental duty or obligation that is central to a role or position.
Example: "As a project manager, ensuring effective communication among team members is a core responsibility."
Alternatives: "a key duty" or "a primary obligation."
Oversight is a core responsibility of my job as a member of the Senate.
Union leaders, like political leaders, are charged with a core responsibility that isn't negotiable: the task of leading.
Of course, plenty of people don't think that guaranteeing affordable health insurance is a core responsibility of government.
The question is whether it represents the future of healthcare, which traditionally, has been a core responsibility of national governments.
Maintaining peace and security is a core responsibility of the United Nations, and any allegation of wrongdoing by peacekeepers is taken extremely seriously.
Regulation of immigration and borders is, of course, a core responsibility of the executive under our system of separation of powers.
But I think the main thing is to have a firm belief about the core responsibility of solving these challenges, and not delegating that to someone else.
Recognizing that this challenge will remain for the medium to long term, a refocus on NATO's core responsibility should be the aim of the September NATO summit in Wales.
Most of all, their core responsibility is to ensure that an idea becomes a reality.
President Obama himself last August described preventing such genocides as a "core" moral responsibility and national security interest of the United States.
The administration justifies the elimination by saying that the work of the cultural agencies isn't a core federal responsibility, and that most are supported with private donations.
