This gap is directly opposite the rhodo, but about three to four feet
away from the rhodo itself. John, the owner has said something about
putting lime in with the roots of the beech when planting, to help to
"counteract" the rhodo, which he says killed the beech hedge at this
point.

Before you consider adding any lime, it might be worth buying a soil testing
kit and testing the soil acidity. You'll also need some distilled water (not
tap water) to add to the soil/tester to obtain accurate results. Only then can
you judge whether it's necessary to add any lime.

Secondly, if the roots are a problem it may be worthwhile digging a narrow
trench down between the Rhodie and the Beech hedge to cut off and roots that
may have spread from the Rhodie and killed the hedge off, though I'm not sure
whether this is actually an issue.

Hope this helps.

Dave.