It depends on the spell.
In general, my reading is that the spell effect comes into existence in the specified area, and any subsequent alterations to the area don't change that.
But, it depends on the spell description. Some spells are explicit that the effect in the area is ongoing and directional, in which case an appropriate piece of cover should exclude that part of the area.
For example, gust of wind creates "a line of strong wind 60 feet long and 10 feet wide" that "blasts from you". It seems obvious that if a wizard throws a wall of stone into the path of the wind, it should stop the wind. You wouldn't have the wind blasting from the enemy caster, then somehow going through the wall to continue to fill the previously defined area.
Similarly, a moonbeam spell "shines down", and I would certainly expect that tossing an opaque horizontal surface would create a shadow that's protected from the beam.
But on the other hand, there's no reason to think that putting a wall through a patch of entangle or grease stops the plants from being big and enchanted or the floor from being slippery.
I think the area continues to be the area of the spell unless the spell specifically calls out or implies that something is being continually emitted from the origin point, which isn't a great answer, but I don't think there's a single rule that makes sense in all cases. Certainly "put a bucket over the origin" would be absurd and potentially trivializing for a lot of magic.