Hi David,
Based on my best knowledge (which is sometimes very poor...

) most of garden furniture is made from polypropylene which is a problematic material form gluing point of view. Though it has a solvent but this chemical is definitely carcinogen and the solution is not applicable as a glow.
A less dangerous home solution could be “welding†with a small gas torch but you have to be very careful – melted polypropylene can cause ugly ulcer on your skin. However I’m afraid is not a good solution for you as the problem is just a hinge.
A couple of years ago I repaired a similar chair. The broken part was the pin and following some drilling exercise I applied a screw and a nut to substitute the damaged part.
Kind regards,
Ferenc