What is the best way to eliminate weeds in mulch?

We purchased a home and the previous owner maintained a beautiful
backyard. We are moving in soon and would like to keep it that
way I do not have much a green thumb yet, but I am trying to get
up to speed as quickly as possible although there is much to learn.

Walking through the backyard the other day I noticed there were many
little weeds growing in the mulch. I say in the mulch because the
previous owner said he had a weed barrier under the mulch. I assume
weed seeds landed on the mulch and are beginning to sprout. I would
prefer not to pick them by hand because there are thousands of them
littered throughout the mulch.

And in the future is it a reasonable idea to apply pre-emergent on top
of the mulch to prevent the problem?

We also have a crushed granite path that is suffering similar
symptoms...I assume this should be treated/prevented in a manner
consistent to the mulch?

What are some solutions to this problem?

Thanks in advance for your expertise.