I would recommend removing the plastic also as it is horrid for your soil. Ground cover would probably be your best option. So, my idea that would work for your budget and help your soil is this:
1. Buy some ground cover, there are many to choose from. Find one that is "hardy" in your
growing zone.
2. Remove the plastic!! To prevent
weeds after removing the plastic, put down a couple of layers of newspaper or brown paper bags. These should be shredded and used similar to mulch. However, it would be easier to
plant your ground cover in it. As the paper begins to degrade, it will add to your soil, but by the time it completely degrades, your ground cover should be thriving.
I have also seen people use several layers of aquarium gravel...but...I don't know if that is something I would personally do. Best of luck.
Ali