"refer the matter" is correct and usable in written English. You can use this phrase when you would like to direct someone to a particular thing or case. For example, you might say, "I can't make a decision on this issue, so please refer the matter to the manager.".
Speaker Nancy Pelosi asked the attorney general to refer the matter to a federal prosecutor.
Miliband declined to refer the matter to the Press Complaints Commission.
The alternative to the ruling would have been to refer the matter to the Congress.
But the investigators decided not to refer the matter to prosecutors.
The DWP, A4e said, had subsequently felt the need to refer the matter for criminal investigation.
They could refer the matter to the Justice Department for prosecution.
The Electoral Act also allows the AEC itself to refer the matter to the court.
Ludwig does not simply clarify my doubts with English writing, it enlightens my writing with new possibilities
Simone Ivan Conte
Software Engineer at Adobe, UK