Weeping forms often hold onto their old leaves. And they also get caught up on the dense network of branches and twigs. You can remove them or not as you wish. They should have no impact on budding or current leaf development.

If the leaves are persisting because of another issue, as some diseases will cause, that's another can of worms altogether. But if that's the case, you will also have no healthy bud development on those branches. And healthy buds should be clearly evident now and generally showing some color.