There will always be some backlash in a normal ACME leadscrew.
The Z lock must be engaged before making any cuts with your machine. The leadscrew will move Z up and down, but will not hold the spindle against the cutting forces (since the only opposing force really is just gravity and the weight of the head).
Just completely loosen the lock screw before making a change to Z, lock it back down and then make your cut. If you’re converting this rig to CNC you’ll have to address the lash in Z, but this is normal for a typical hobby benchtop manual machine.