LP ? Undersized spindle hole.

@slaw is exactly right -- use the Xacto and take it slowly taking care to remove the shavings as you go. Getting a drill anywhere near an LP is a recipe for disaster -- one slip and the whole thing is ruined, not to mention it is nigh impossible to ensure you have the LP centered.

btw the same method can be used to move offcenter LPs by expanding the center hole (assuming you use some sort of clamp to stop the LP wandering)

Regarding spindle diameters they do vary which is why both my alignment protractor (AS SmartTractor) and clamp (Durand) come with interchangeable spindle adaptors

Just push in a flat blade screwdriver thats bigger than the hole and rotate once. Blow off the chaf if any.

Medium grit sand paper on a tapered dowel will take care of this quickly and precisely if you keep the dowel at 90 degrees to the album. . 

Here's Clearaudio's solution.

When I worked in a machine shop (early '70s) I used something called a "ream". Metal like a drill bit but made to make a hole exactly the perfect size. Took off minimal material...