Plant ivy at the bottom of the hedge and train it up just cut it with the hedge to control it.
Are you thinking of growing something the street-side of the hedge or the side facing your home?
Which way does the house face and what sort of soil do you have in the garden. Also, how high is the hedge?
The only thing I can think of that might establish between the roots of a hedge (apart from ivy) is Vinca major (periwinkle), which would also spread out sideways and need controlling.
You could consider putting some short posts and wire netting to the front of the hedge and letting the hedge grow through it so that it's more or less hidden - that would stop the rubbish collecting in the base of the hedge.