How exactly do you know that the power is going back on?
If the TV is still plugged into one recepticle and simply plugging the AC tester causes the TV to come back on ---- then you need to immediately investigate repairing and replacing the duplex recepticle. It is likely that the wiring in the box is using the quick connect type connection where the wires poke into a hole in the back of the outlet. One wire must be loose and making an intermittent connection. This could be caused by a wire that is improperly installed or the internal string contact that is supposed to retain the wire is broken. Simply installing the AC tester into the other outlet is apparently moving the contact metal work inside the recepticle just enough to cause it to touch the loose wire.
Do note that a loose wire like this is unsafe and could lead to a fire - so please investigate immediately - for the sake of yourself, your family and your dwelling.
The answer is not a cut and dry one in your case. The troubleshooting is a process. You will need an understanding of a multiple light parallel circuit. You will also need a proximity type voltage tester and possibly a VOM.
Assuming the voltage feed starts at the switch (not always the case, but normal) you will need to verify input voltage at the line side of the switch with the switch in the off position. If you have voltage there, turn the switch on and verify voltage on the load side of the switch. At this point, a VOM is handy to test voltage across the hot and neutral and/or ground. If this looks good, proceed to the closest light fixture, and with the switch on, test the center hot tab in each fixture with the no-touch tester. In your case I would think you may not see any voltage based on your question. This could mean that you have an open neutral, an open hot, or at worse, a shorted hot to neutral/ground.
Since there are several possible reasons for your condition, it would take a lot of tutorial to explain every possible scenario. Assuming you do not see voltage at the fixtures, the basic technique for troubleshooting will be to start at the last known verified voltage point, then follow the wiring and check all connections in the junction boxes. Visually check the bulb sockets for broken or shorted metal tabs. This should be done with the power off at first, looking for obvious loose or disconnected wires, then with the power on using your voltage tester. Since this condition occurred after you adjusted the height of the sockets in the fixtures, I suspect tension on some wire may have pulled a wire out of a socket base or out of a wirenut in the j-box. Unfortunately, most can/pot lights have a built in j-box attached to the top of the fixture. This necessitates dropping the fixture down below the ceiling to access the j-box, or getting access from above. (attic).
Basically, you are following the circuit looking for an open or short, just like following a hose, looking for a water leak or stoppage.
This is not a hard job, but extreme caution must be taken when testing energized circuits. If you do not have good electrical skills, the proper test equipment, or a logical understanding of switched paralleled circuits, then this job is better left to a pro.
Maybe one of my buddies here on SE can add a good graphic showing this type of circuit and the test points. A simple line drawing showing the switch and junction points would be a great edit.
Best Answer
If you have a voltage reading between hot and ground but not hot and neutral, it means you have an open neutral somewhere between the bulb and the panel. Since you were playing around in the fixture, that'd be the first place I'd check. Remove all of the wirenuts and check the quality of the connections, both from the line as well as the socket. If the fixture has a transformer there will be another set of connections to check (primary and secondary sides).