Repairing a mech shoulder

Shoulder and Hip actuator critical hits cause such serious structural damage that they can't simply be replaced, the entire limb needs to be rebuilt. You will need to purchase a replacement location of the appropriate size and type (eg a 20 ton standard left arm or 40 ton endo steel right leg etc). To make the repair you will need to remove the parts of the limb that are salvageable and scrap the parts that are beyond repair.

In the repair window if you right click the mech's name in the list of units needing repair you will see an option to switch to salvage mode, in this mode your techs will be removing parts from the unit instead of attempting repairs. To later switch the unit back to repair mode you right click it again and the repair option will be in the menu. While in salvage mode you will need to remove all surviving armor and components from the limb with the destroyed hip or shoulder actuator. Once all the surviving parts have been removed you can right click on the location in the list of parts and select scrap. After the location has been scraped (and you have switch the unit back to repair mode) you will be able to replace the location just like a location that has been destroyed in battle and reinstalled or replace the components in the location.

While MekHQ gives the warning against repairing components in a location with this damage sometimes you will have reason to do so. For example if a replacement limb is going to take two weeks to deliver but you have a desperate need for that mech in battle before then you could repair the damage that is able to fixed without replacement and accept the penalty for the damaged hip/shoulder.