Final Cut Pro - how can I rejoin clips that were originally split from the same clip?

If you are referring to a through edit, where there is an edit in a clip that essentially is a split point in continuous material, then select the edit point by clicking on it and then go to the menu Mark > Join Clips - or use the keyboard shortcut of the delete key.

MtD

It will only work if you have the edit point selected, as already noted:

It will work on any clip that shows a dotted line. It doesn’t matter how long ago or even if the system has been rebooted. If any change is made to the cut point then the through edit cannot be deleted. Please show a screenshot of what you’re seeing.

As Tom says, a through edit has the dashed vertical line, as shown in the screen shot I posted earlier. If there's no dashed line, then the clips do not have identical properties and can't be joined-- something was changed on one of them (transformed, cropped, etc.)

To illustrate Meg The Dog's excellent advice about joining a through edit, from Cut clips in two in Final Cut Pro for Mac - Apple Support

Can you post a screenshot of what you see after you blade the clip. Are you selecting the edit point?

The delete key is the one next to =. On my MBP is marked delete.