OK, first of all, plainly this is a mess. The fact that there is a white wire hooked up to a black wire without any recolouring of it alone says that (1) there's some amateurish stuff going on here, and (2) you cannot trust any of the white wires to actually be neutral until you prove that they are. So proceed cautiously, and take notes as you go.
There are a lot of questions here but a good one is:
What's the easiest way to track a specific wire to see how it flows without ripping out the walls?
Start by obtaining a long three-prong extension cord. Plug it into an outlet that is on, and verify with a voltmeter that there is voltage between the hot and neutral, there is voltage between the hot and ground, and no voltage between the neutral and ground. You now have an extension cord which you can use to provide a known ground and a known hot.
Next, turn off the power to everything that you're going to be working on.
Next, put a piece of tape with a number on every wire that you're going to be working on, and start making notes about the color and location of each numbered wire end, and what they're hooked up to now. Take photos. You want to be able to put this back together the way you found it later. Number the switches as well.
Next, carefully get all the wire ends exposed but not touching anything, and turn the power back on. Use your known ground and your voltmeter to determine which wires are hot under every combination of switches. Remember, some of the whites are hot.
Turn the power back off again.
Test everything again to make sure that no, really, the power is off.
Now test everything again against the known hot to find out which wires are neutrals that run back to the panel. Again, test everything against all combinations of switches, remembering that at least one mistake has been made in this wiring so far; there might be more. Interrupting a neutral with a switch is almost always wrong, but I have found plenty of interrupted neutrals in old houses.
Now unplug your extension cord from the wall and get a flashlight. Rig yourself up a continuity tester out of the extension cord and the flashlight. (Of course if your voltmeter has a continuity tester, use it instead of jury-rigging one out of a flashlight.)
Suppose you suspect that wire ends labeled 2 and 9 are the same wire. Attach end 2 to the extension cord ground. Run the extension cord to the other. Now wire the extension cord to the battery, the battery to the light, and the light to end 9. Did the light go on? Then your hypothesis is confirmed. Did it not go on? Try every combination of switches. Again, record your observations.
Keep doing that until you have enough observations to form a consistent theory of how the wiring runs through the walls. Draw a diagram.
Once you're there, you should be able to solve your problem. If you cannot, then you'll be able to post a question here with a lot more information.
Also, take this opportunity to ensure that every wire is correctly coloured, by putting tape on miscoloured or confusing wires. Also, if there are oddities such as a box that has hot wires from two different breakers, make a note of that for the next guy.
It's either a loose connection, a bad switch, or a poltergeist.
- Turn power off to the circuit.
- Verify power is off.
- Remove the cover plate from each switch.
- Remove the screws holding the switches to the box (should be two, one top, one bottom).
- Pull the switches out a bit.
- Inspect the wiring and connections, looking for loose wires, charred/burnt/melted parts, and any other damage.
- Pull the lights down, and inspect the wiring there as well.
If you don't find anything, try replacing the switches. If there are 3 switches controlling the circuit, then you'll need two 3-way switches and one 4-way switch.
If you don't feel comfortable doing this type of work, contact a local licensed Electrician.
Best Answer
You can branch off the 3-way switched circuit of the can lights to the a new single pole switch that then in turn switches the reading light can. However in this scenario the reading light can light will always be OFF when the rest of the can lights are OFF.
There will not be a straightforward way to mix the two 3-ways with a 1-way to control the added can light so that the the added one can be ON when all the other can lights are OFF. The simplest is probably going to be to tap into a live circuit power source and then just independently control the reading light with that one way switch.
There are some types of smart switch setups that can be configured to get the can lights to work somewhat together and still allow independent control of the reading light but you would have to carefully research the various technologies and find the one suitable for your application. Be prepared to spend a fair amount of cash as the typical smart switches cost upwards from 8 to 10 times what a common manual switch would cost.