Rather than simply archiving unwanted messages, configure Google Sync to delete messages on your iPhone:
1. Go to m.google.com/sync. Here you’ll see a list of your iOS devices configured to use Gmail. Choose your iPhone.
2. Select the option to enable Delete Email as Trash. Then tap Save.
TIP: Google Sync features are only available via Microsoft Exchange. Google Sync via Microsoft Exchange includes such features as “push,” which means that the email will be delivered to your iOS device as soon as it arrives.