Core2Duo p-socket CPUs upgradbale?

Your are correct the p in the CPU identifier implied a socket option was possible.

In the case of this system: MacBook 2.13 GHz Core 2 Duo 13" (White-09). The CPU is Intel's 2.13 GHz Core 2 Duo (P7450).

Now lets look a bit deeper here:

Here's the Intel ARC specs for the CPU: Intel Core2 Duo Processor P7450 (3M Cache, 2.13 GHz, 1066 MHz FSB). We see Sockets Supported - PGA478.

Now what is a PGA (Pin Gate Array): A socket which has pins that push into the solder pads on the bottom the the chip (BGA). Here's the chip carrier that was designed Socket P. It was found the socket had issues.

The chip was also designed to be used in a soldered configuration. Here's the BGA (Ball Gate Array) info: Ball grid array.

OK with all of this you have a better idea on the basics.

Apple did not use socketed CPU's on any of its MacBook or MacBook Pro systems. Mostly due to space as the extra height would have forced Apple to make a thicker system. Also Apple did not trust the sockets reliability which is why it had a very sort life span even with other hardware companies.

OK, lets say the system had a socket what would it gain you? You could not install anything other than a Core2Duo and you would also need to tune the FSB clock if you put in a different chip than what the system was intended for. Review the ARC DB and you will see your options are very limited here a few .xx GHz difference between the chips (basically very little difference).

I would focus on upping the Ram to the max and swapping out the HD for a SSHD to gain the deeper caching the SSD on the drive offers. I would go with a Seagate unit here as your systems SATA I/O port speed is SATA I (1.5 Gb/s).