WHAT TO PLANT ADJACENT TO CORNISH HEDGE?

My garden is bordered on east and west faces by a Cornish hedge - comprising mainly grass, cow-parsely, bramble, ivy, hawthorn, oak. The problem is I would like to plant something interesting adjacent to them but fear any planting/bedding will be overrun by the weed/grass etc in the hedges - can anyone suggest some suitable planting ideas please. Thanks in anticipation.
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