How do I delete the circled vertex, whilst maintaining the outer geometry?
There must be an option to split a line into two where another line intersects it.

You may enable snap to edges and move your circled vertices along the edge (‘g’ twice’). The vertice will stick to the intersection. Than you subdivide the transversal edge and merge the vertices.

Thanks Angleo,
I subdivided the line as that dot wasn’t a vertex but a dot drawn on the pic by me, as RPaladin suggested then pressed g twice , after enabling the snap to edges.