First, your illustrations are Mad Awesome. You could illustrate electrical books. Literally. You might even talk to Mike Holt or others doing electrical docs.
You still have some knowledge gaps, so I'd school up some more. For a guy as smart as you, knowledge is cheap.
If you are good at visual, stay with that. Buy a variety-pack of electrical tape colors, and a couple feet of 12/3 cable because it's a cheap way to get a variety of wire colors for pigtails. 12 gauge is the universal donor size, it is acceptable on any common 120v circuit up to 20 amp breaker. 14ga is only allowed on 15A breakers/with 14ga wire.
First, permanently wrap (tag) the white wire of cable C with red tape. From your comments elsewhere that there is only one cable going to the switch, that is a switch loop. Also open up the switch box and wrap the other end of that same white wire.
Next, permanently wrap (tag) the black wire of cable A with red tape. Since the switch is a switch loop, this cable is the only possible way the lights could possibly be receiving (switched) power.
Now grab your receptacle and some stripped Romex and sit at a convenient workbench. Put 6" pigtails of wire as follows. Use the screw terminals or screw-and-clamp if you have that type. Avoid backstabs (they're not reliable) and never use 12AWG on a backstab.
- Ground terminal: a bare (or green) wire.
- screw 1: a white wire.
- screw 2: nothing, but if the tab between 1 and 2 has been broken, a white wire.
- screw 3: a black wire.
- screw 4: nothing, but if the tab between 3 and 4 is broken, a black wire.
Ready?
Splice all same colors together.
See, what I did was color-code all the wires to their function rather than the default colors of /2 cable. The switch loop has only hot (black) and switched-hot (red). The wire to the lights needs switched-hot (red) and real neutral (white).
In new work, they commonly use red for the switched hot, because the law now requires neutral in switch loops (for smart switches). So they run some /3 cable up there.
I was taught to feed the hot from the top. It could be the bottom or top. Get a noncontact voltage tester to see which one is hot with the switch in the off position. Trial and error will work here as you only have 2 choices and it will not hurt anything to use the wrong one your outlet just won’t work when the light switch is off. Other than that yes White to White, Grounds all tied together.
Make sure your new wire is sized right, 20A breaker 12 gauge wire, 15 A breaker 14 gauge wire. (you can use 12 gauge wire on a 15 amp breaker just not the other way around).
Best Answer
The proper thing to do here would be to free the wire from that clip and then wire nut that bare copper wire with two additional bare copper wire pigtails. Green wire nut would be ideal. One of the pigtails will go to the green screw on the outlet assembly. The other bare wire pigtail needs to be connected to the box. It is unknown if the clip can be used again so barring any judgement I would look to see if the electrical box has a pre-threaded hole in the back. The threaded hole is intended to accommodate a green box grounding screw. Get one and use it to attach the second bare pigtail to the back of the box. The picture below shows what appears to be the pre-threaded grounding hole in your box.