vschneck Jiggling the cable near the port makes the power go on and off.
Did you check to make sure that the barrel connector is a 2.1mm/5.5 mm one? A 2.5mm/5.5mm connector will plug into the same 12V camera jack, but the center contact (+ve) will be loose and intermittent like you describe.
The easiest way to find put is to try to plug the existing power cable into a 2.5/5.5 jack. If it can seat all the way into the test jack, the connector is the wrong one to use with ZWO cameras.
I keep known 2.1mm and 2.5mm plugs and jacks. Before using any, I test it against my reference connectors.
Next, if it is truly a 2.1mm/5.5mm plug, unless the actual jack in the camera appears mechanically loose (cold solder joints), it is otherwise hard to destroy the jack end of an axial connector. More likely, it is the plug end (which is lucky for you) that has become worn out. Try a different connector (i.e., cable).
Chen