On Wednesday, 16 September 2015 11:29:24 UTC+1, News wrote:

Thank you both for your comments, which make perfect sense. I must
admit that my first thought was turf, as that will help smother existing
seeds.

It is worth making a good job of levelling and smoothing off before laying turf, partly because it makes a smoother lawn, and partly because it makes for more even rooting down of the new turf.

The old advice used to be to rotovate, roll, and rake until like a billiard table. Lot of work.