At least three rivers called the Clarence River:

Australia

The Clarence River flows through Grafton, New South Wales, Australia. Its source is Rivertree and its mouth is at Yamba.

Canada/USA

The Clarence River rises in the Yukon Territory of Canada and crosses back and forth across the border several times into the U.S. state of Alaska.

New Zealand

The Clarence River is located in Marlborough, in the South Island of New Zealand.