This is partially a problem with how POP3 email works and partially the Samsung email client not communicating properly with the email server. The email client is not correctly determining what mail is new and what has been already downloaded.
But if you set your phone email client to delete mail on the server (and it actually works, which is iffy) then when you check mail on your office desktop PC the mail you downloaded first to the phone will not be available, it's not there for download anymore. Conversely, if you have the PC set to delete mail on the server, mail downloaded first to the PC will not be available to the phone. But, if you set it to leave mail on the server, it keeps coming back.
So, what do you do? You can...
1. Try another email client on the Samsung S5 and see if that stops the resurfacing emails.
2. Set both the PC and S5 to always leave mail on the server. This is OK if #1 worked and you don't want to miss any mail on either the phone or PC. But you have to clear old mail from the server now and then to avoid filling up the space allowed on the server.
3. Convert the POP3 account to IMAP if possible. This is by far the best solution.
Android since v1.0. Linux user since 2001.