Hello ryerson48,
You'll first want to determine exactly where the issue might be coming from. If it's only with Gmail, check their website and see if you make any changes on the device they reflect here as that is where the computer and the iPhone will retrieve that information.
If the deletions aren't syncing to the Gmail website at all, you'll want to remove the affected account and add it again to one of the devices: Add and remove email accounts on iPhone - Apple Support
1. Go to Settings
> Mail > Accounts.
2. Tap the email account you want to remove, then do one of the following:
* If you’re removing an iCloud email account: Tap iCloud, tap iCloud Mail, then turn off “Use on this iPhone.”
* If you’re removing another email account: Turn off Mail.
* Note: To remove the email account from all apps on your iPhone, tap Delete Account.
To add additional email accounts, do the following:
1. Go to Settings
> Mail > Accounts > Add Account.
2. Do one of the following:
* Tap an email service—for example, iCloud or Microsoft Exchange—then enter your email account information.
* Tap Other, tap Add Mail Account, then enter your email account information.
Take care.