There has been quite a bit of work on this at both the ecosystem and physiological levels. Previous efforts to identify specific limits for nutrient concentrations at the ecosystem-scale, however, have been strongly questioned in the classic literature. For example, the effects of high nutrient concentrations (i.e., algal productivity) should be countered by corresponding increases in the rates of herbivory, and reefs have been shown to exist in environments around the world with various nutrient regimes. The greatest concern regarding nutrients occurs when herbivore populations are unable to control highly productive algal communities due to fishing or disease, and possibly when nutrients are unnaturally increased at rates too high for herbivores to respond. The direct effects of nutrients on the coral holobiont (coral-zooxanthellae-bacteria consortium) and coral growth is discussed in another body of literature that I am less familiar with.
An example of the ecological debate (e.g., in the Caribbean) can be found in these articles:
In fact, there is an enormous literature on this important subject. Nutrients definitely affect reefs, and nutrient pollution is probably the major reason reefs are gone along so many coastlines.
It is important to understand that "nutrients" may refer not only to inorganic ions such as phosphate and nitrate, but to a spectrum of dissolved substances. Nutrient increases have been connected with increases in bioerosion, declines in growth rate, increases in disease and changes in coral species composition.
I suggest reading the work of David Baker, Kiho Kim, Max Wisshak and Christine Schoenberg. I have attached some relevant papers.