Sheffield-Carlisle Off Peak return is £50.20 (Valid Anytime)
This is valid via S&C or via Newcastle (and possibly via Preston)
Splitting at York/Newcastle doesn't reduce the figure.
If you can go via the S&C both ways, you can split at Leeds
Sheffield-Leeds (Route Not Doncaster) £11.20 (or £8.80 after 0930)
Leeds-Carlisle (Route Appleby) £27.80
Total = £39 or £36.60
Freedom of North East 4 in 8 days is £75, only £24.80 more than Sheffield-Carlisle, and is valid after 0845 and is worth getting if you are planning more trips in the area within 8 days. At £18.75 per day it's good value if so.