Coming back to the original thread title, and remembering we are in the Speculative discussion sub forum, I would suggest that the 3rd Hope Valley express should be a TPE service from Manchester Piccadilly to Lincoln service calling at New Mills Central and Dore. The resulting services would be:
1tph Manchester Picccadilly - Lincoln (New Mills Central, Dore, Sheffield, Worksop, Retford, Gainsborough Lea Road, Saxilby, Lincoln)
1tph Sheffield - Gainsborough Central (reinstate the pre covid service)
2tph Leeds - Nottingham (the current Northern Leeds - Lincoln becomes a second Nottingham per hour)
This reinstates 2tph on the Lincoln route, gives Lincoln a service to Manchester, doubles one of Northern's busiest routes (Leeds-Nottingham) and at the loss of Leeds - Lincoln, which lets face it, is much quicker via Doncaster, so really the loss is Barnsley and Meadowhall to stations on the Lincoln line, which yes is a loss, but I would argue is significantly outweighed by the gain on the other routes.