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The whole idea of successful plantings of ground cover is that you choose the sort of plants that will spread and therefore cover the ground.
As you have covered the ground with landscaping membrane even if you make holes in it to plant plants, they won't be able to spread either by sending out root runners or by seed.
Perhaps you should remove the membrane in the areas where you want groundcover plants?
Yep, if you want ground cover, you need to remove the membrane where you want anything to grow - it's job, after all, is stopping things from growing. Whip the relevant bits off, wait a couple of weeks to see if weeds spring up, spray them off or dig them up when they do, and then you can plant. I'd suggest you tackle an area at a time and go steady - you'll get an idea of how things are doing then before you commit yourself. If you can see whether its sunny/shady, wet/dry soil, and preferably acid/alkaline I could give you a list of plants to consider. I'm big on ground cover myself, particularly under trees.