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Yes, for different pages with older content we have recommended to hide them from Google. This specific page, however, could is highly relevant to their audience and we want their audience to be able to find it so we don't just want to hide it. We have looked for solutions on the help forum but haven't found anything relevant to this issue. Does anyone know if there is a way to tell Google that our website is safe to avoid this problem in general? Or is it just a matter of finding other relevant keywords and synonyms that are currently not being used to see if that will help Google better understand the topic of this page? Or could there be another problem that we're missing?

Just wait, things will clear up by themselfs if that was in fact what caused the drop. SE' will want to see stability over a period of time, who knows what that is, but one would think it would involve them crawling you a fair few times to come to the conclusion that you are now stable.