Be very careful when opening up PSUs like that - although they may be unplugged, there may still be plenty of charge in the on-board capacitors (which should ideally be discharged safely).
Which plug do you want to replace (the part where you mention a fragile centre pin), as I'm not sure I understand from your diagram. The highlighted part (red circle) looks like a cable that is soldered directly to the PCB.
If it's the end with the DC plug on it, it's probably easier to snip the cable, strip it back and solder a new DC plug on the end and heatshrink the cables (rather than replace the whole cable).