What better example than the decaying remains of a group of Hosta rectifolia foliage to remind us that a good edge to a lawn needs an annual renovation? Continuing to provide a splash of autumnal beauty, these leaves soon shrivel and disintegrate with the wet inclement weather we will now experience. These leaves are low on lignin and the cellulose content decomposes rapidly.
As for the lawn edge renovation, choose a dry day. Cut out the damaged section of the turf edge, ideally 20mm thick so that the root growth binds the turf together making handling easier. Make the piece you lift as long as manageable. Turn so that the good edge now forms the border edge. Lightly cultivate the exposed soil to a fine tilth. This will allow roots to grow down and bind the turf into the soil.
Back fill the exposed soil and sow with grass seed. Alternatively re turf the whole patch if enough turf is available. If dry, water in.
Alternatively you could widen the width of the border or lift and divide the offending material. Replant with a fresh eye to the mature dimensions of the group.
Whatever you decide; act now. If not you will spend the winter months looking out of the window at the situation and sighing with despair that you did not act sooner!