The pith of store-bought cut kabocha squash tends to harbor mold growth. If it's a whole kabocha, it doesn't spoil easily, since the inside is not exposed to air.
Accordingly, this is a method for storing cut kabocha squash.
Carefully scrape out the seeds and pith with a spoon. Scrape by forcefully carving them out.
Put it in a plastic bag, then store it in the vegetable compartment of your refrigerator. That's it. It lasted for a little over a week this way.
Addendum: Thoroughly scrape out the pith with a spoon until you can't scrape any more. If it's still moist after removing the pith.
Let it sit out of the refrigerator for half a day to dry the surface, then store it in a plastic bag.
