Also cannot help with 1924, but an LMSR timetable for Summer 1928 shows:
From Liverpool Exchange dep.:
09:02 (SX) to Hull; via Bolton, Bury Knowsley St. & Rochdale;
combined at Rochdale with Manchester Victoria to York & Scarborough service. York arr 12:25
09:02 (SO) to York & Hull, arr. York 1225
10:55 to York & Newcastle; called Manchester Vic., Wakefield & Normanton; York. arr. 13:42
14:45 to Manchester Vic, connection to York, arr. 17:56; train also conveyed Manchester to Burnley portion as far as Rochdale
15:55 to Manchester Vic., connection to York, arr. 1941
18:35 to York, arr. 22:10 (semi-fast via Wakefield Kirkgate)
From Liverpool Lime Street dep.:
09:30 to Scarborough; York arr. 12:35
All other trains from Lime St. required a change at Leeds.
Return from York
09:55 to Manchester Victoria, connection to Liverpool Exchange (13:20)
12:05 to Manchester Victoria, coonection to Liverpool Exchange (15:20)
15:50 (14:45 ex- Scarborough) to Man. Vic; connection to L'pool Ex. (19:15)
19:07 (16:50 ex- Newcastle) to Liverpool Exchange (22:08 SO, or 22:20 SX)
13:12 SO (12:05 ex- Scarborough) to Liverpool Lime Street (16:40)
15:30 SX (14:20 ex-Scarborough) to Liverpool Lime Street (19:00)
--------------------------------------------------------------------------------------------------------------------