New cases are either "New" from the standpoint of A.) the date they were added as a case (ie. entry date); or B.) from the standpoint of the date of their first assistance (ie. assistance date) (which can also be relative to the assistance category being used to filter the report.)
B.) In an "Assistance" report, the Report Summary: New Case Count = the number of cases that have never been assisted before the start date of the report. See screenshot here.
The same applies to the Report Summary: New Household count = the number of households that have never been assisted before the start date of the report. See screenshot here.
If a household is split, and a new household with a new street address is formed, it will only be considered as a "New Household" if it meets the criterion of "a household never assisted before the start date of the assistance report". Creating a "new" street address does not automatically qualify it as a "new" household.