Normally, if you download messages with a POP account, the messages are removed from server unless user specifies to not do that. Maybe more information is needed.

Thanks for replying. My settings are to leave message on the server for a few days, say 7. Thus it would be convenient to have a shortcut to delete messages from the server without modifying the general setting. The filters have the option "delete from server" but I do not know how to make a menu item from it.

I am not a shortcut key user, so can't give much advice. There is an addon, tbkeys-lite, that may be of use. Here is a list of all thunderbird shortcuts: https://support.mozilla.org/en-US/kb/keyboard-shortcuts-thunderbird?redirectslug=keyboard-shortcuts&redirectlocale=en-US#w_writing-messages

Consider setting up the accounts as IMAP instead of POP on each computer so all actions are synced between the devices.

The two computers share the same accounts but their behaviours are distinct. IMAP is not a solution for me.