What is Causing a Cyclic 1kHz Noise Signal in Audio Recording?
"The UCA202 uses the ubiquitous TI PCM2902 integrated USB DAC chip"

So this a "Full speed" USB device, and the USB data is being sent at 12 MHz, but it doesn't send data continuously; it sends it in short bursts every 1 ms, which means it has a frame rate of 1 kHz.  This 1 kHz noise finds its way into everything and it's really annoying when you're designing USB audio products. 

Sometimes it becomes much more audible because of a ground loop and you can improve it by lifting a ground somewhere, like unplugging your laptop from the wall, or connecting to a balanced input on your monitor instead of an unbalanced one, or using an audio transformer to isolate one thing from another.


  There is nothing plugged into the UCA202.


What if you short out the inputs instead of leaving them floating?  They're RCA inputs so they might not have normals that short themselves when not connected to anything.