Sounds like it could be a ' mouthlamp ', as used by gasfitters years ago for soldering joints on lead piping.
The cylindrical part would be filled with methalated spirit, this would seep up to the exposed part of the wick, or wadding as you say. A 2ft. long piece of small bore rubber tube would be connected to the fine tapered tubing.
The exposed part of the wick would be lit, to produce a constant flame, the gasfitter would then blow down the end of the rubber tube, and, as the other end of the fine tapered tubing, which should be copper ,by the way, bends over the wick, this produces a fine flame point which is ideal for melting solder into the prepared joint on the lead pipe.
I still have mine in the garage, still in working order. I would post a photo of it, but don't know how to do that.
Hope this helps.