A CORE World IS a World that you have assigned a Governor (Leader) to. You could make a size 1 Colony World into a CORE World if you wanted, but it would normally/usually a very bad/worthless idea. You might want to Core a fairly small World if it had a special resource available that you wanted to develop, but usually they do not any to develop.
A Core World is one where you can give orders to build Districts or Special Buildings on the Tiles, including Terraforming any Tiles. However, really small size Worlds often have no or very few (1 or 2) tiles that be Terraformed even with the high level Techs. I do not know if there might be some way to do something with those really small Worlds, with some really High level Techs/Events/Ideologies/etc. Maybe, but only develop them (turn them into Core Worlds) if you are sure it can be done.