How do I repair this?

This has been popping up on some video games I play similar to Egg Inc. and PG3D and it wont let me connect with video games utilizing my Gmail. Ive already made progress within the sport I took this screenshot in and if i had been to delete it, i might lose 2 years value of progress. Hopefully somebody can inform me methods to repair this to have the ability to use my Google Play Video games gmail on video games to have the ability to save my progress. Im getting a brand new telephone and that i don't know if it'll switch completely all of my stuff. If anybody can discover a answer to this that may be nice.