I found the below on a different Mac forum. Perhaps some of these instructions can be used to remove its rear stand, expose its TORX screws and allow one to "tighten the hinge up" themselves.
If under 1 year warranty or under extended AppleCare support, I'd "call it in" and get them to tighten up the hinge screw. Assuming one can live without their computer for a few days.
Good luck....
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I found my VESA Mount Adapter manual, but it doesn't have instructions for removal, but perhaps reversing the installation instructions will do the trick. Here's the relevant parts:
Turn your iMac so that the back is facing you, then rotate the top forward until it won't go any further.
Release the latch by inserting the *access card* (thick plastic credit card ??) in the space directly above the top of the stand. The access card should go into the space at a slight upward angle for about 3/4 of an inch. If it goes further in, you've missed the latch, so try again.
Rotate your iMac forward until the 8 screws on top of the stand are visible and the stand is locked in place. Important: Keep your iMac upright until the stand is locked in place.
Hold the sides of your iMac and carefully lay it down so that the screen is resting on a soft towel or cloth on a flat surface and the stand hangs over the edge of your work surface.
Use the Torx tool to remove the 8 screws that attach the stand to the back of your iMac.
I'm surprised about two things: that there aren't explicit removal instructions in the manual, and that Apple doesn't have a pdf of this manual in their online support documentation. That is sloppy support on Apple's part.
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