The problem you describe is generally called "cable-borne noise" or "mechanical cable noise", and is due to the lack of an isolation loop.
Disconnect the 3.5mm plug. If the cable is routed through the beams of the stand, extricate the cable from the assembly.
Use electrical tape to affix the cable to one side (not the bottom) of the lower beam, 2 or 3 cm (about an inch) from the bolt. Be sure to leave enough slack to allow changing the microphone angle, without the cable and pop filter gooseneck touching.
Make a 4 inch minimum diameter loop, and affix the other end to the other side of the beam and about 5cm (a couple of inches) farther back. No parts of the loop should touch.
If you like, you may route the remaining cable through the descending beam.
When I make isolation loops, I often wrap a bit of plastic foam (not bubble wrap), about 3mm (1/8 inch) thick, around the cable areas taped to the mechanism. Foam with adhesive on one side to adhere to the cable is best.