Sadly not. In the corporate world, if an activity doesn't have a timesheet task code associated, it doesn't exist.
Clearing e-mails doesn't have a task code, so effeectively it's a few hours unbillable on top of your working hours each day.
Then don't do it. If that stops other people getting their jobs done, that's not your problem. Trying to fix dysfunctional organization problems with "I will work harder" doesn't get you appreciated, it gets you a ticket to the glue factory. (Reference from Animal Farm by George Orwell.)