Up until the other day on my iPhone/iPad I always used the IMAP protocol to access my Gmail accounts instead of Exchange because of the way Google handled deleting emails. For Exchange users Google would archive an email anytime you hit the trash icon, and if you really wanted to delete the message you’d then have to go into your “All Mail” label and delete it from there. It was a tedious two-step process, and that is still what they have in place by default. The good news is that you can switch this behavior now.
Google added an option to the m.google.com/sync site so that you can have emails trashed when you hit the delete button instead of archiving the emails. Basically all you have to do is visit the m.google.com/sync site on your mobile device, select your device from the list, and check the box to Enable “Delete Email As Trash” for this device.
That may sound pretty easy, but if you have a bunch of accounts it can be quite a pain since you have to login to each one to change the setting. If you have a lot of accounts I recommend using a desktop browser where you can change the user agent to an iOS device, which will make it go a little faster to login and logout of all the accounts to enable the delete instead of archive option.