When I told Itunes to sync my email accounts between Macbook Pro and Iphone5 I began to have problems getting email to work on the Iphone. Each time I accessed emails I was asked to provide my password again and again with no end. I reset the password with Google so I know it have it right.
My usual solution is to delete my mail account and recreate it. When I go to Settings>Email, Contacts, Calendars and click on my Gmail account, I'm bounced out of Settings and back to the Home Screen. I'd like to reinstall the Mail app but suppose that requires reinstalling IOS6. I want to be able to retreive all my apps (some not from App Store) and all data, but what I read about Restore from Itunes backup, I'll still loose a lot.