What you describe is classic symptoms of backlash, which is typically caused by one or more of the following, in no particular order:
1) Backlash in the ballscrew/ballnut
2) Inadequate pre-load on the ballscrew bearings
3) Worn ballscrew bearings
4) Ballnut not properly secured to its mounting block
5) Loose or broken shaft coupling between the ballscrew and the drive motor
6) Loose gibs
In your case, it would be the Y axis that is at fault. The only way to fix it is to check each of the above, and figure out exactly where the problem is.
Regards,
Ray L.