To answer your question, a brief overview of how e-mail works is required..
There are two forms of email: POP and IMAP. These are the acronyms that the services use, and it's how email providers get you the email.
POP works quite simply, and the name is fitting. When you use POP, you "pop" the emails off the server and to your device/browser/etc. Because you're effectively downloading the email to whatever device you're viewing them on, when you delete an e-mail it's gone off the server. POP is characterized by two-way communication and syncing: when you view/delete/compose an email on your device, that change is then sent back to the e-mail server and reflects on all of your devices.
IMAP works a bit differently. When you open your e-mail client that uses IMAP (like, in this instance Gmail) you're just downloading a copy of the e-mails from the server to your device. When you make any change on an email (reading, etc) it's device-specific, and won't reflect on the server.
Bottom line is that due to the way the mail protocol works with Gmail, unless you're using the web client, you can't truly delete emails. For more information, check here.
Edited by UnloosedCake, 30 September 2017 - 04:39 PM.