What does Benjamin Franklin mean by saying: My testimony in Whitefield's favour ought to have the more weight, as we had no religious connection? What is the exact meaning of testominy over here?

Yes, Franklin is noting that his support is based primarily on Whitefield's character, nothing more. The fact that Whitefield was a preacher, and that he had a large following, had nothing to do with it. Franklin supported Whitefield because of who he was.... not what he did. In context, testimony might be defined as recommendation.