If you actually mean can you put the solar cover on the pool surface and run the filter and pump too. BTW, the filter does not move the water, the PUMP moves the water. The pump is the only piece of equipment that you would be used to run th ewater thru the filter. The pump is like the heart of the equipment pad.
Then yes you can do both. The solar cover when installed as it should be – on the water – has no signifigence with regard to the filter ( the large tank).
The purpose of running the pump either with the cover on or off would be to move or circulate the water. With a solar cover on the surface of the water – only the top two feet of the water is warmed. The rest of the pool remains cooler. Circulating the water would move that warm water and mix it with the cooler water. Think about it.