It's not clear how to check the background job result for App Edit Costing Run.
SAP S/4HANA Cloud
Reproducing the Issue
Run steps in App Edit Costing Run with background processing.
It's possible to check the job execution status via App Application Jobs.
There will be spool request created in the column Result.
You can check the result to see whether there are any messages created.
CK40N, Selection, Costing, foreground, error message , KBA , CO-PC-PCP , Product Cost Planning , Problem