I can't tell for sure, but it definitely looks like your pad is gone. If you have an exacto knife you can just gently scrape away the solder mask, find the trace, and solder to the trace.

You lost your pad. This usually happens if you have old PCBs which were exposed to some air humidity over time. The pad heats up and the moisture evaporates between the layers, separating the pad and the PCB base material. More heat increases the likelihood of that happening.

That part is unsolderable now. You need to connect at a different place and reconnect any interrupted traces or glue the pad back to the board, if you can.

Game controllers in general are very hard to repair since they contain large copper layers to withstand forces from the controls. It’s very common to loose pads or lift THT holes.

That said, most often nothing is ever beyond repair. It’s usually just a matter of skills and tools which are available.

As others have said, your pad is gone. If you're struggling to expose the trace needed to solder, it might be easier to find out which one of those test points corresponds to the battery terminal and solder a wire to that instead.

Here's a great resource I used to repair an Xbox one controller after destroying the pads in an attempt to replace an analogue stick.

To understand the original traces on each side of a circuit board, you can put a powerful flashlivht on the back to see through.

I can't say if the hole is too wide and makes a short circuit with another close track.

Do you use flux ?
Have you cleaned the track with water on a coton tip ? (To remove battery leak) What kind of soldering wire do you use ? Shitty soldering iron does not stick.

I had the same issue with a circuit. Gently sanding and using flux solved my issue. But it was an ugly soldering. I am not proud of it. You are not alone.