I need to attach a lamp to a bed headboard, it came with a really shitty plastic clamp that keeps falling over. The lamp has a 1cm diameter metal rod as its attachment point at the base, so basically I want a metal bracket that can screw into the headboard and provides a vertical cylindrical sheath to drop the lamp into. Here's a rough mspaint drawing of what I'm imagining, front view and top view (with screws visible).
What kind of terms should I be searching for to find this? I can't figure out how to describe this in a way that will let me buy one.