Controlling Cypress Aphid
This is the most common cause of brown patches on established conifer hedges including Leylandii caused by large greyish coloured greenfly (although by the time the damage is evident the aphids are often long gone). Sometimes the shed skins of the aphids can be seen, other symptoms include a sticky black sooty mould that develops on the sugary liquid excreted by the aphids although this is very much a secondary infection and again, will appear before the browning of the foliage is seen.
Cypress Aphid can be controlled with a suitable insecticide spray which will help to limit the damage. If caught early enough then the brown patches should grow back in over time, this can be hastened by tying in neighbouring healthy shoots to help cover the affected area. This may also be helped by mulching the hedge and applying a feed, ideally in the spring to strengthen the hedge and help it to recover.