Reworking WAGO 2060 on MCPCB
They are broken already.  Just break them some more and make your life easier:

Use diagonal cutters to clip away the damaged connector.

Then, you can unsolder and remove each pin individually.  

Use a solder sucker or solder wick to clear the holes after the pins are out.  

Once that is done, you can insert and solder the new connectors.



That is the method we used to replace bad through hole parts when I worked in a factory building 2-way radios.  It is less damaging to the PCB than trying to unsolder  a broken part.

It is also what we did if we suspected a bad part.  It was cheaper (and faster) to assume the part bad, clip it out, and replace it than to try to figure out if it was really the problem.  

I mean, replace a 5cent part in five minutes or spend an hour diddling with it.  Clip out and replace wins, hands down.