Country location adjoining hedge between my neighbours.
Hedge runs approx 30ft by 6ft in height x 2ft thick.
Hedge gets an annual prune, but over time has become wider in one section about 10ft to the point it leans in to the garden. As it's an evergreen of some sort I don't want to aggressively cut that section back in case the leaf never returns. Attach photos showing problem area and close-up of hedge leaf.
Any ideas where it could be cut back.

All hedges should have the shape you choose. Each year cut it back to that shape. If it has gotten to 10 feet wide the annual trimming has not taken it back enough.Because winter is coming you might wait until spring when it starts growing again. Keep it fertilized to make it strong.

Cut a branch and show it to a local nursery or you county extension agent to have it identified. You could also take close up photos and post them in the 'Plant ID' forum. Different bushes need different care.