No pictures unfortunately, but I'll try and describe as best I can.
I sometimes get spots that are like a hard lump, sometimes with pus or a blackhead, but then they seem to develop a core, which is of a semi-hard white-ish matter. A bit like a blackhead but more solid. This core is attached at a sort of root, and the original lump of the spot only seems to heal when pulled out at the root with tweezers. It's definitely not milia, but nobody seems to know what it is, apart from my dad who has had a similar sort of thing.
Any ideas? The "core" is usually 2-3mm long.