It would potentially be sensible to transfer one or two services to EMR, but I can't think off the top of my head which
There's a slippery slope of routes that you could transfer, but it's a case of where you draw the line.
EMR are taking the Barton - Cleethorpes stopper from Northern, so already have a 75mph DMU operation on the south bank of the Humber.
The truncated Scunthorpe - Doncaster stopper (i.e. doesn't run through to Meadowhall/ Sheffield) arguably has more in common with EMR - EMR also run Doncaster - Lincoln and Cleethorpes/ Grimsby - Lincoln, so you could give them the Scunthorpe service.
Personally, I'd be fine with EMR taking the Sheffield - Gainsborough stopper and the Sheffield - Lincoln semi-fast (which could be better co-ordinated with the Robin Hood services at Worksop), but that would remove the Meadowhall link.
What after that though? IMHO, it'd be good to have one TOC running the two "fast" Hope Valley services, but if you give the Manchester - Cleethorpes route to EMR then do you throw in the Northern routes on the Sheffield - Doncaster corridor too? That'd meant that Sheffield was the centre of a franchise, rather than being on the fringes of Northern/ TPE/ EMR.
The problem with privatisation is that it's a lot harder to engineer these kind of swaps, since you'd have to negotiate with all of the various franchises, so the only thing I could see happening would splitting Northern into two (not giving the other bits to anyone else)
Would a separate Tyne/Tees franchise be a good idea? I don't think there are any Northern-run through services to the area from the south, it's a fairly well-separated operation (although I'm happy to be corrected). The Newcastle-Carlisle service would be run by this new franchise.
The only real connection to the rest of the network is that most of the South Yorkshire Pacers have been based at Heaton (so periodically have to scamper up/down the ECML at 75mph to rotate them - that must be quite an experience for the poor driver!). Other than that, it's pretty much separate (if there are any 158s that swap with S&C diagrams at Carlisle then that's pretty minor in the grand scheme of things).
The only real problem I've got is whether it'd be big enough to stand independently or whether there's an argument for putting it in with a longer distance franchise that already operate in Tyneside/ Wearside/ Teesside (i.e. TPE or LNER) - it'd have more interaction with them than it would with the remaining bits of Northern.
There's then the problem that, if you're going to have a separate TPE franchise, it needs to have a certain critical mass - the main Leeds - Huddersfield - Manchester route isn't enough in itself so you need to bundle something else in with it - it used to be the Manchester Airport - Blackpool/ Cumbria services - maybe it'd work with the Heaton services (which are operationally separate, but there's maybe some benefit to promoting them and managing them as part of another "half franchise" worth of routes?)