I want to add insulation to the cavity in order to reduce noise etc.
Insulation won't do much to reduce noise. However it will reduce heat loss, so probably still worthwhile for that reason.
Or would it be easier to add some sort of soundproof plaster board to the existing wall
That is more like it. To reduce sound, you need mass, i.e. heavy solid items between you and the source of the noise. Ideally, the plasterboard should be fixed to framework which is independent from the wall structure, otherwise vibration (and therefore noise) will be transferred from the wall to the new boards.
Sockets will mean a hole in the wall through which sound will pass, so best to remove the socket completely and put it on another wall.
One solution could be:
remove the plaster and laths on your side of the wall
insulate between the vertical supports
attach soundblock plasterboard to cover the whole wall
tape and plaster the joints between boards
attach resilient bars horiontally to the new wall
fix another layer of soundblock plasterboard to the bars
tape and skim the plasterboard.
That lot should only be about 2 inches thick, so minimal loss of space in the room. If you double up the plasterboard layers this will improve the performance, but obviously mean the wall is thicker.
Mieral wool type insulation will be better than foam boards, as the wool types have more mass.
It's also essential that the new plasterboard wall is sealed at all the edges - floor, ceiling and the adjoining walls. A flexible mastic sealant will ensure cracks do not develop later.
If you don't know what resilient bars are, google will reveal all.