The picture of the line types and the accompanying comment helps to clear this up:
they are "Travel" lines, but why material is printed during travel?
Yes, the dark blue is travel. But if material is deposited during them, then that is not printing, that is material oozing out of the nozzle - which is called stringing most often. Usually, this is an issue with retraction: your length is possibly a little short. However, retraction also only is turned on for moves over a given length, so there might be no retraction happening. And, in Cura support structure generally is printed without retraction due to toggle
Limit Support Retractions. Toggling that off can help.
You might also turn on one of the two Avoid modes in the movement (
Avoid Printed Parts when Traveling &
Avoid Supports When Traveling) to get the movement paths in a way so they don't go over the model and prevent any stringing/oozing onto it or knocking off delicate parts. Possibly increasing retraction a little might help, and so might
Z-hop to try to "rip" the strings.