Ok 2000 miles are roughly 3200 km. Now with an earth sized planet of roughly 6300 km radius you have arctan(1600/6380)*2=28,15 degrees covered. This means that roughly 30/180*24 h= 4h are without sunlight. That’s roughly the time a solar eclipse can last. This of the time you won’t have sunlight. At that it will be at an lesser angle reducing the energy per area. Generally I’d assume that this will come with a lowered temperature in that specific area.
But as pointed out in the prior thread depending on the kind of object hovering it might also be that sunlight is captured in the morning if the material is reflective or the material emits absorbed light like a black body in all directions or is something in between. So there is also the possibility that the disc functions as cover because energy absorbed in the morning and evening by the earth is absorbed and emitted or reflected during the time the sun stand above your disc.
This possible thermal coupling between the earth and the disc makes it quite hard to guess the weather situation just based on the size. Let’s say the disk would be made of glass than it might function like a green house. But this might still be the case if the disk is black while absorbing and remitting radiation above the surface.
So I think you have 2 main choices here. If it’s a reflective material it’s possible that the temperature changes are more drastic and there is a time of cool down when the sun is blocked. If it’s more absorbent it might heat up the covered continent somewhat more even though there are 4 hours less of direct high sun light. In addition to that will the disk absorb light from the sun longer than the earth does ..so it might slowly release halve of the absorbed energy back to earth.
Later is however more speculative on my side. After all the radiation is directed in all direction and not let through like in a green house. So cooling could also be be happening as with the mirror but the temperature rise in the morning evening might be not so noticeable due to the fact that the material re-emits some of the absorbed energy instead reflecting it.
All of this might result in changing weather situations. Like a greenhouse it might be more warm an wet or it might become quite hot in the morning and evening when the light is reflected, while during mid day the temperature drops. The result it rain might be coming down due to less water absorbed in the air when it gets cooler.
Than there is also a coupling with the surrounding due to convection ect. Is the surrounding less warm for example due to prolonged shadowing and lesser thermal coupling. In that case i suspect that is a chance for forming high and low pressure areas going for the outside to the inside. In case of a reflective material this might be different so. It’s simply very hard to say without knowing what the thermal coupling would be like.
Having a more noticeable effect would mean to increase the size even more.