If you do have to prune anything at all.. Why throw away useful offcuts after pruning or send them away for the council to recycle?
Do it yourself by first separating the leaves from the woody stems.
The leaves can be used as a mulch somewhere in the garden, out of site if you prefer, or in a hide-away makeshift compost heap behind an evergreen shrub.
The woody stems can be tied up in bundles and dried in the shed, to be used as kindle for BBQ fires during the New Moon, or in the winter indoors, if you are lucky enough to have access to a working fireplace.
Bits from roses, pyracantha and bramble with thorns on them are best kept and dried separately to be discarded for the fire first, unless you want to encounter them again at a later stage with your bear hands!
Stacks of wood piled up in the garden to dry over time becomes habitat for a multitude of useful insects, like centipedes who will keep snails and slugs population down. Wrens will also nest in undisturbed piles of wood over-run by bramble and Ivy.