This tree is in my front yard. It has a itchy powdery stuff all over it.
Stays green all year and buds white flower things at the ends in spring.

Closer view needed of first one to tell. And lopping of branches should be discontinued as that destroys the beauty and purpose of a weeping specimen.

Second is leather-leaf viburnum. This also has been getting mal-pruned, would be much more attractive and effective left to branch to the ground, form a dome-like shape.

I hope this close-up view works for you. Thanks!

Weeping mulberry.
M.alba.
There are 'fruitless' male clones, like Chaparral, and fruiting female selections, like 'Teas Weeping', among others.