Peter Hickman held down eighth after a fighting performance on the Tyco Honda, ahead of Michael Rutter, while PR Kawasaki's Simon Andrews scored his first EVO class win of the season with a run to tenth.
The recovering Kirkham and Brookes collected 11th and 13th, the pair split by Martin Jessopp, while Scott Smart and Dan Linfoot completed the points' paying positions.
In the overall standings, Byrne leads Hopkins now by 31 points, with Hill and Laverty holding down third and fourth. The absent Stuart Easton maintains his place inside the top six with fifth, while Ryuichi Kiyonari occupies sixth.
However, the final spots remain closely fought with Kiyonari, Hickman, Kirkham, Rutter, Mason, Baz and Brookes all separated by a mere 22 points.