Join Date: May 2004
Location: Wilmington, DE
Bikes: 1984 Trek 660, 2003 Specialized Hardrock, 2004 LOOK KG386i, 2005 Iron Horse Warrior Expert, 2009 Pedal Force CX1, 2014 Islabikes CNOC 14 (son's)
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It sounds like the threads in the plate have been stripped. They are formed by directly tapping the steel plate which has been punched in a way to create a short cylinder on the surface (common method of creating substantial threads in sheet metal). Since the screw won't back out on its own, you'll probably have to drill the head off to release the cleat.
Once you get the cleat off, you'll need to figure out how to remove the old plate because you'll have to replace it or at least remove it to modify it. Shimano doesn't have any information online for their shoes so you may be stuck calling/emailing them. On my Shimano MTB shoes, the insole comes out and then a piece of plastic tape (that you install to "seal" the cleat holes after installation) needs to be removed to access the plate. Newer road shoes are probably different.
If you can't find a replacement plate, I would suggest grinding off the cylinder on the sheet metal and replacing the threads by using a thin nut, or jam nut, in the appropriate size. You'll need to hold the nut from spinning with a wrench when tightening your cleats but at least your shoes will be useable.