How do I wire an additional recessed light to an existing switched circuit?
There probably is no neutral at the switch box. A switch with "one set of wires" (I am guessing you mean one cable - pictures or a clear description would help me be accurate) is almost certainly a "switch loop." Under newer codes that should be a 3-wire (red-black-white-ground - ground is not counted in naming the cable, so "3-wire") cable to supply a neutral to the switch box, but under older codes a 2-wire black-white-ground cable was often used.

One wire (should be black) will be the hot to the switch. The other (should be red on the 3-wire, or on 2-wire the white wire which SHOULD have red tape or paint applied to it) will be the switched hot, from the switch, back to the first fixture. The neutral for the circuit will be at the fixture in this case.

(Pardon - North American color codes - other parts of the world do have other conventions, involving blue, brown, etc...)

Likely what you need to do is move the new light's wire to one of the other lights, not the switch.

Edit, Add - what you probably are doing is putting the one new light in series with all the other lights in parallel between it and neutral. If you remove the bulbs in all the other lights, odds are good that the new one will go out. If you remove most of them, you'll probably find that the new fixture gets dimmer, and the remaining lights on the circuit start to glow a bit...all the lights should be in parallel (from hot to neutral) but what you have now is hot, the one light, an intermediate connection from the one light to all the other lights, and through those other lights to neutral.