What is a pivot ?

The pivot is the end of the gear arbor.  It's kinda like an axle.  The pivot, which is steel,  rotates in a hole in the clock plate which is brass.  This is also where the clock oil goes.  over time, dust and dirt sticks to the oil, and now you have an abrasive mix that wears the hole in the plate larger.  If you think dust isn't abrasive, get a tiny bit in your eye and let me know how that feels.
Occasionally the grit will get imbedded into the softer brass and instead of the hole wearing out, the steel pivot will actually get scored. If worn badly enough like this one, all that can be done is to replace it.

First it is cut off flush to the end of the arbor.  Then in a small lathe a hole is backdrilled into the arbor at true center.

What I do next  is to take the next larger size drill bit and press it into the hole.  You don't want to solder it because the heat is liable to soften the steel. The excess is then cut to length and shaped.

All thats left to do is fit it to the hole in the plate.