What kind of "weeds" do you have?
Are these "weeds" growing in the soil or in the mulch?

I believe they are growing in the soil. I will take a photo of the weeds as I have no idea what kind they are. It is a different kind of weed every season and there are two or three different kinds each time.

"Weeds" are a fact of life, they will be there because birds eat the seeds of many and the wind carries the seeds of many and those seeds land, germinate, and grow. A good thick mulch can help control them, but it will not totally eliminate them. The thickness of the mulch is also important because the usually 2 inch thick mulch will do little to keep any "weed" seeds in the soil from germinating and growing. A good mulch needs to be at least 4 inches thick, but "weed" sees can still get dropped on that mulch, germinate, and grow, however those "weeds" are very easily removed.
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atlcharm - I'm in the process of dealing with about 1/2 acre that has gone unmaintained for 15+ years. I don't have all of the answers, but I can give some advice from my experience:

1) Pull by hand. I tried a string trimmer and the stuff was back in 2 weeks. Anything pulled by hand has stayed gone for more than 2 months with only a few pop-ups. Hand pulling is the cheapest and generally the most effective weed control for an area covered in mulch. Also, the first time you hit a milkweed seed-pod with a weed whacker and watch all those seeds fly around your property, you become a believer in hand weeding :)

2) Weed seeds in your mulch will sprout. Whether they are the ones already there this year or the ones that blow in or get pooped out by the birds next year. No big deal to handle since the mulch is much less dense than soil (hopefully).

3) Since I'm a generally broke, I'm not going to pay for 1/3 of an acre of plastic. I have been gathering as much cardboard as I can lay my hands on and putting a double layer of it underneath the mulch that I have put down on a couple of beds. It keeps the weeds down very nicely and should be suitably decomposed in a year or so when I get ready to plant things in those beds. The rest of the property is going to get a few inches of free mulch from the town facility and I'll weed it by hand.

4) Another thing about the plastic is that it doesn't allow water to penetrate. Since this area is on a steep slope, you might want to go with a water-permeable landscaping cloth rather than plastic (if you decide to go with anything at all). This would help to prevent the water from collecting on top of the plastic and washing your mulch down the slope.

Sounds like my project!
Steep slope once with rug juniper.
The junipers were dieing at the top to mid way of the slopes, I tried to put vinca vine to cover the woody dead stuff, no luck. Mulched over and had the same issues you had with weeds popping thru.
Ended up spending most of the early spring tearing out the old dead wood, by hand. What a chore that was.
once the ground was bare I had a better time controling the weed infestations from soil disturbance. I had to spray round up 2-3 times to get them under control, as I have trees nearby where birds roost.
Later I started to replant with ground covers and daylilies.
The ajuga is working very well for erosion and is quickly filling in, as is the creeping phlox. For the areas I am working on I first put down a thick layer of newspaper , then put the mulch on top of that. those areas have minimal weeds there. I did this in sections because the slope is huge also.
I was able to reclaim one section of the rug juniper, and to get those weeds out of there I allowed the weeds to get higher than the groundcover and dense, then I sprayed the tops of the weeds, that also worked well. I will be spraying again for the summer weeds that grew.
keep in mind that different weeds have different germination times.
An excellent site to find you weed emergence is:
WWW. weedalert.com
I hope this helps you.