Salix Intergra 'Hakuro Nishki'

I have at least seven of them. You might be able to see 2 of them, near an arch in my garden. I usually just cut the unwanted bits at the bottom off, quite neatly near the trunk, and do nothing else. Everything seems fine.

You need to remove them as close to the trunk as possible. The trunk is not the same plant as the variegated bit on top. Any part left will grow again from dormant buds on them. Do not damage the trunk itself though. In future, keep your eye on the trunk and any buds which appear may be rubbed off when tiny, it saves a lot of trouble.