Hello,
The cross jet, being perpendicular to the beam direction will keep the spatter away from the focussing optics but will not provide protection to the weld pool against oxidation from atmospheric air. Normally compressed air is used for cross-jet and an inert gas, also called cover gas (Helium, Nitrogen, Argon) for weld pool protection. I am still not clear the way you are bringing the cover gas to the weld area.
Are you trying to improve the weld seam appearance (spatter around the weld seam) or back spatter towards to the focussing optics?
If the weld appearance is not good (smooth and regular) may be you have too much of power. Here comes the beam quality into the picture. If the beam quality is not good, you will have to use extra power to reach the desired weld depth but at the same point the material at the surface overheats and causes spatter. Using Helium as cover gas may help, it cools the weld pool, limits the plasma and improves the weld depth. If the component is small, this becomes more tricky because the total thermal mass is low.
Hope this helps you.