Yes that movement is normal and will not affect the finish of your cut. It is due to the usual backlash between the cross slide feed screw and the nut.
Cause of poor cutting could be any of more than a dozen other things: eg slack in headstock bearings, blunt or chipped tool, tool height, cutting speed, feed rate, poor piece of material, worn bed, gib strip adjustment on topslide, cross slide or main carriage, etc etc.
Edited By Hopper on 10/06/2023 10:10:12