This might not be the answer, but for reference here is a problem that can occur if the plumbing is not installed properly.
If you look at Fig. 3, this is what a proper drain looks like. You'll notice the orange line represents the water level in the system, the water levels out in the trap preventing sewer gases from entering the sink drain.
In Fig. 1, you can see what will happen if the drain line has to go up before meeting the main drain line. Again the orange line represents the water level in the system, and as you can see the water level is much higher in the drain.
Fig. 2 shows what could happen if the sink was installed lower than the rise in the drain line. The water would never fully drain from the sink, since it would require extra pressure in the system to push the water up the drain.
![enter image description here](https://i.stack.imgur.com/Ws9y4.png)
I wouldn't guess this is the issue in this case; since as you can see, if you opened the trap in this case you would have more water drain than what is typically in the trap.
In a perfect world Fig.1 and 2 would never pass inspection, but as we all know not all plumbing is properly installed and/or inspected. So this could possibly be the case in this situation (given the plumbing was installed by an amateur plumber), but it is doubtful. It would be more likely caused by a clog in the line.
You could check if the remaining piece of pivot rod that you have in hand has magnetic properties (i.e. a magnet will stick to it). If so you can probably be successful in retrieving the broken part from the trap using a tool similar to those pictured below:
![enter image description here](https://i.stack.imgur.com/HpjDo.png)
![enter image description here](https://i.stack.imgur.com/30hQO.png)
Magnetic reach tools can be purchased online, hardware stores or big box outlets. In addition to the type shown above that have the flexible shaft there is another type that has the shaft made out of a soft aluminum wire that can bend slightly as the thing is pushed into the drain line. This latter type is typically a lower cost than the types pictured.
Best Answer
The usual fix for this would be to extend the part of the pipe from the wall at the back of the cabinet out the amount needed to match up for the new sink drain location. In some instances there is a larger diameter pipe in the wall that the rear piece slides into and is secured with a compression washer and a clamp nut. The horizontal slide joint makes it much easier to adjust the P-trap joint.
In your situation due to the glued connections at the wall it can be more work to resolve this issue. Especially since it does not appear that there is much allowance to just cut and glue in new fittings. You will likely need to cut off the existing P-trap elbow from the rear wall extension. There is a small space just behind the elbow where this cut can be made. What you would be cutting there is a short piece of the pipe that is about 2 1/4 inches long that was glued in to join the P-trap elbow and the next fitting that extends out of the wall.
Once the elbow is cut off the next step is to hopefully be able to remove the remaining stub of pipe that is glued into the fitting from the wall. There are specialty rotary cutter tools that are made for carving out pipe stubs in existing fittings so that a new pipe extension can be glued in its place. This is one example of such a tool that is powered by an heavy duty electric drill (often one sized as a 1/2" drill).
(Picture Source: https://www.homedepot.com/p/JSC-2-in-Socket-Saver-J44200/302575481?MERCH=REC-_-PIPHorizontal2_rr-_-302575482-_-302575481-_-N)
The other ugly alternative that you have in a situation like this is to cut into the drywall to open up the wall enough so that you can access the drain pipe where it is possible to cut where you can then use a coupler fitting and new pipe and fittings that are measured out to meet the new sink location. Subsequent repair of the drywall makes a much larger project of the repair such that the cost of trying the tool shown above is a very tidy alternative.