Privet (Ligustrum) is very easy to take cuttings of, both softwood and
hardwood root very easily, you will need to be a little patient before they
are large enough to fill the gaps in your established hedge, but once it
gets its roots down it will grow on vigorously. I would recommend starting
some cuttings off in autumn or spring in a nursery row before moving them to
their final planting position once they have a decent root system. When you
do plant them in the gaps make sure the planting hole is enriched with
compost so they get a good start as the soil around the hedge will probably
be exhausted.