Broken connector on a1990 motherboard?

Well not good😣

It looks like the traces have been torn off when the ZIF connector was extracted! But you also have a problem on the other side as well as that ZIF connector is missing its compression latch!

So you'll need to get two connectors at least (best to buy three) one to carefully snap off the latch to then snap onto the current connector. They do tend to fly off in the process which is why you want a second. As to the logic board side this is going to take some deep soldering skills and trace patching to get it working.

As to what's this cable (you might need a new one if damaged) it is the communication between the batteries BMC logic and the main logic boards SMC battery charging control logic. It's needed for the computer to work properly.

You'll likely need to get the systems password and will need to have the previous owner remove it from his iCloud account so the system is free and clear. He may need to login via the TouchID to clear it as well.