While your USB phone charger has enough power to keep everything running, it was not designed with sensitive analog electronics in mind. The small shift in output voltage when you energize the relay has no effect on your digital electronics because everything is still in spec, but your analog electronics use that voltage as a reference and small changes in that voltage coupled with the large gain you are likely using with the HX711 changes your scaling enough that you’re seeing changes in your output.
When using the USB port for power, the Vusb input is tied to the +5V rail on the Nano via a schottky diode (see the Nano’s schematic here: https://www.arduino.cc/en/uploads/Main/ … .2-SCH.pdf ) which will have a small voltage drop associated with it. If you connect Vusb to +5V, you are eliminating that drop but you are still fully relying on the stability of your USB power source for everything connected to your project. There is a chance that this will improve your operation but no guarantee. This is why it is critical to isolate the power going to your relay from the HX711. It’s a good design practice to keep power electronics isolated from analog and lower power digital circuitry specifically for the problem you are encountering here.
-Bill