How to block-delete Webmail in-box entries?

[#1]

As I very seldom use Webmail unless abroad, my in-box gets clogged-up with duplicates of my Plusnet e-mails.  Is there a way to manually 'Delete All', or tweak the settings so that incoming mails are automatically deleted after a set period?

[#2]

It all depends on what client you use and how it's configured.
It appears your using the POP server  (which downloads a copy to your system), and the client is set to leave messages on the server.
You should be able to modify that to limit the number of days they're kept, and possibly to track deletes.
Alternatively,  reconfigure your client to use IMAP, which acts with the server rather downloading locally.
This will mean that webmail (which uses IMAP) sees exactly the same emails as your client.