Various carpet types have different absorption coefficients throughout the frequency range. A thick pile carpet will absorb more sound than a thin one. What makes even more difference though is the type of padding you use. For best acoustic properties and a more even absorption extending down to lower frequencies, use a hair-felt padding. This is kinda an old-fashioned padding, most of the stuff today is foam of some kind, but if you can find it, the hair-felt would be the best. As your second choice, use a foam with open cells - like foam rubber. A padding with closed cells or a combination of closed and open cells like sponge rubber won't be quite as good.
One of the earlier suggestions was to keep the wood floors and use an area rug - which could also be a good option if you do other acoustic treatments in the room. The problem with carpet is that it absorbs frequencies above 1kHz pretty well, but doesn't work very well especially below 500Hz, so you get some uneven absorption across the frequency range. You can even it out by adding the additional plywood to the floor like you were suggesting, and perhaps also adding a few bass traps. Or as suggested before, you could keep the wood floors, and add an area rug between the speakers and your listening chair to catch the first reflection from the floor, and then add the acoustic panels to the side walls, and on the wall behind the speakers. Without the wall to wall carpet, you will need more panels than if you had the carpet. Either catching the first reflection from floors, walls and ceiling can really focus the sound and improve the soundstage.
One of the earlier suggestions was to keep the wood floors and use an area rug - which could also be a good option if you do other acoustic treatments in the room. The problem with carpet is that it absorbs frequencies above 1kHz pretty well, but doesn't work very well especially below 500Hz, so you get some uneven absorption across the frequency range. You can even it out by adding the additional plywood to the floor like you were suggesting, and perhaps also adding a few bass traps. Or as suggested before, you could keep the wood floors, and add an area rug between the speakers and your listening chair to catch the first reflection from the floor, and then add the acoustic panels to the side walls, and on the wall behind the speakers. Without the wall to wall carpet, you will need more panels than if you had the carpet. Either catching the first reflection from floors, walls and ceiling can really focus the sound and improve the soundstage.