The phrase "advancing the relationship" is correct and usable in written English.
It can be used in contexts where you are discussing the improvement or development of a relationship, whether personal or professional.
Example: "Our goal this year is to focus on advancing the relationship between our two companies to foster better collaboration."
Alternatives: "enhancing the relationship" or "improving the relationship".
Will Xi have the strategic foresight to invest in advancing the relationship with the United States?
Advancing the relationship requires the active engagement of China's next leader — and the American president — to build strategic trust between the two great nations.
He emphasized his point by saying, "we have fought with you in six wars, and we do not share your democratic values," pointing out his disagreement with a former U.S. ambassador on the importance of democratic values in advancing the relationship between the United States and its GCC allies.
Conversely, if the date ended with an awkward hug or an unappreciative romantic gesture, then it is likely that either 1 or both of the daters was not interested in advancing the relationship.
They function, like a shrug of the shoulders, as physical dialogue, advancing the relationships and the drama between the dancers.
To be fair, the best episode among the five sent for review was the fifth, in which the action started to kick into gear, and there were touching scenes advancing the relationships between Dracula and Mina, and Dracula and Renfield.
With regard to family-professional partnership and interaction (research question 1), first of all, this study showed that the qualities identified as those which advance the relationship between families and service providers were persistently lacking (Summers et al., 2005).
In an attempt to advance the relationship between explicit, computational models and physiological data collected during the performance of cognitive tasks, we developed a PDP model of visual word recognition which simulates key results from the ERP reading literature, while simultaneously being able to successfully perform lexical decision a benchmark task for reading models.
