The old school way is to open a terminal and use
mail
at the prompt, then
d
to delete one message or
d1-4
for for four messages.
Then
q
to quit.
If you don't want to delete them unread, just type Enter at the mail prompt and it'll show each message in turn.
Use
man mail
for all the details.
Of course, other more sophisticated mail clients may be available, but that will do it.