That "hinge" is the connection between the governor arm and the wire link to the throttle on the engine. Hence it moves when the engine is running, to keep the engine speed steady at whatever you set it to with the control lever, regardless of the load on the engine. Make sure there is nothing obstructing that linkage.

Some details on your engine. First, it was made on 27 September 2011, in the Briggs plant at Chongqing, China. All Briggs side-valve engines are now made in China: the small ones like yours have been for about ten years.

Here is the Illustrated Parts List for your engine:
http://bsintek.basco.com/BriggsDocumentDisplay/18ktxIXUnfBhU7y.pdf

You can use that to see what parts there are, what they look like, how they connect to each other, and what their Briggs part numbers are.

Here is the Operator's Manual for your engine:
http://bsintek.basco.com/BriggsDocumentDisplay/iorpzNJU0HCjhJvcv4.pdf

You can use that to see how to operate and maintain your engine correctly.