Summary:
How to find out which day of the week the users have entered time against and then segregate the same as Weekend ( For days between Monday To Friday) and Weekends ( For Saturday and Sunday)

Resolution:
It is not possible to track entry date as Weekend/Weekday using in-built reporting functions in Web Timesheet.

Workaround:
To achieve the same, create a custom column in reports using the following code.

IF((MOD((Daydiff((date(1990,9,9)),EntryDate)),7)=0) || (MOD((Daydiff((date(1990,9,8)),EntryDate)),7)=0),"Weekend","Weekday")

This would create a column in reports where:
It would show Weekday for hours entered on weekdays ( Monday To Friday).
It would show "Weekend" for hours entered on weekends ( Saturday/Sunday)