Do native americans have asian ancestry?

A theory exists that Native Americans originally came from Mongolia. Across through Russia into Alaska, down to Canada, US, Mexico, Central America, and South America. It makes sense to me and I believe it.

It's the Bering Strait, baby!

Yes, I'm pretty sure they do.

Except for some Sub-Saharan African peoples, everyone currently alive, has 'Asian' ancestry, to some degree, and through some, or many lineages.

The people who fist inhabited the New World continents, previously inhabited coastal South-Western Europe. They are currently known as Solutreans. Their ancestors migrated Westward from Central Asia (as did all other European peoples).
There were also, at least six later migrations, from people who migrated to the New World from Eastern Asia, and had admixed with the first peoples, to varying extents. These later peoples, also had common distant ancestors in Central Asia.

Of course, all, or most of these people (including Europeans and North Africans, but not all Mongoloids), were hybrids with Neanderthals (Homo neanderthalensis), Subtribes, or Subspecies of these 'other' people, to varying extents and lineages.
The neanderthal people, were a cold-adapted Palearctic species of humans, pre-dating the later, more gracile Homo sapiens.

Yes, they are distantly related to eastern Siberian people.

Very very far back yes