What is the Mandarin term for great godparents? What does my child call my godparents?  Do my godparents become "great godparents" now that I have a child, similar to my parents becoming grandparents?
To most people in English "great godparents" means simply godparents who are very very good at it.  In general, "godparent" means different things to different people.  People especially differ as the the religious connotations.  For many Christians, a godparent has promised to assure that the godchild will grow up properly schooled in the faith, if the parents are unable to.

People who want a Chinese term for godparents have used 乾媽 and 乾爹.  But I think in China if you say someone is your 乾媽 or 乾爹 most people will think you mean that person is your foster parent or step parent.  In the west a godparent might become a foster parent, if the parents die, but usually that does not happen.

By far the most common use of  乾媽 on line is to name 老干妈 chili sauce.  And in Taiwan at least one person has used 乾媽 to mean oranges.  See  http://news.ltn.com.tw/news/society/breakingnews/1593333

So I think if you want to express your idea of godparents in Chinese, you should first be clearer about your idea in English.  If you mean, for example the Catholic Church sense of the term, then you could ask Chinese Catholics how they express it.