That depends on your manual skills.can I solder an extra length of wire of similar thickness to the tail in order to reattach to the pins.
This connection was a copper wire to a solder pin the day it was born. Now you want to know if you can connect a copper wire to a solder pin. What's the difference? A couple of millimeters of copper wire attached to the end of dozens of feet of copper wire? Will your kitchen faucet stop working if the water pipe is 1/8th of an inch longer?
Yes. It might change the probability of a long life. Any sort of repair is difficult to do to the highest possible standards unless you're good at it. I could do that. Can you?Will this impact the performance of the transformer in any way?
You have three options: Try to fix it, buy a new transformer (and attach it properly so it doesn't get jerked off the board when people drop it), or give up and throw it all in the trash bin. Pick one.