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some bikes have one wire coming from the bulb and that's it. as the casing earths to the frame and makes the complete circuit.
so yours might do that too. you may find it earths to the threaded part, although its likely a circuit board so may not do that.
but if thats the case you could strip the wire back and sandwich it in between a couple of washers or to the frame.
its just something to try, not necessarily a fix but will get you on the road until the replacement turns up.
It’s plastic and all glued together. Looking inside as much as possible, the black wire seems to be connected to the light strip and the red one to a small metal piece inside which makes me think that red is earth right? Will it work for now without an earth and only the black wire connected while I order new lights or wait for a replacement from eBay?