It seems a shame that nothing seems to be being done with the Cadishead line; whilst the Cadishead Viaduct over the MSC does need expensive repairs doing to it, the rest of the line is fairly intact, as are the two junctions.
In conjunction with the Midland line from Heaton Mersey to New Mills, the reopening of the Cadishead line could provide a useful 'Manchester cut-off' route for the speeding up of routes like Liverpool-Sheffield, avoiding the need to go through Manchester.
Are there any plans to reopen the route through Cadishead?
I'm assuming by 'Midland line' you mean the line that leaves the Altrincham-Stockport line about halfway along, and runs South of Stockport to link to the existing Manchester-Sheffield route just past Hazelgrove?
There are a few merits: The Deansgate-Piccadilly corridor is very congested and anything that could take passengers off that corridor would be welcome. You'd also speed up Liverpool-Sheffield times not just by avoiding Manchester, but also by avoiding the slow section between Stockport and Hazelgrove. But on the flip side, as others have mentioned, it's hard to see that there'd be enough through Liverpool/Warrington to Sheffield/beyond traffic to justify all that work. And although you may speed up a few journeys, you haven't actually enabled any journeys that can't already be made fairly easily. And the number of people you'd remove from central Manchester would be quite small. Seems to me like a lot of investment for relatively little gain.
More seriously in my view, you'd be running non-stop along lines that go through some heavily populated areas in south Manchester. Wouldn't it be better to use those lines for local services that can serve the residents of those areas?
I suggest a different approach: If money is available for this kind of investment in the area, then surely a far bigger priority has to be upgrading the Altrincham-Stockport line with some new stations so you can run a decent Manchester-Stockport-(new stations)-Altrincham metro service. And of course, once you've done that, it's going to be hard to mix in fast Liverpool-Sheffield services along that line, as your proposal requires.
A second priority would surely be the Western link to Manchester Airport, connecting Altrincham to the airport, ideally via the new HS2 station (and presumably, thereby extending the aforementioned Manchester-Altrincham services).
Only after you've done that would there possibly be a half-decent case for the Cadishead line - but that would probably also be most suited to local services - most likely to give a Liverpool-Warrington-Altrincham-HS2 station-Airport link. That would seriously improve connectivity, enabling a lot of new journeys that are not really possible today. And it would likely remove a much larger number of passengers from the Deansgate-Piccadilly corridor (basically, most people from Liverpool/Warrington who want to get to HS2 or to the airport) so helping congestion in central Manchester.