The on-page grader lists the pages that rank in the top fifty for a keyword in one of your campaigns. In other words if you are seeing a phrase, even if the associated page is graded an F, it is because that phrase brought up that page in the top 50 search results.
In general the rule is one phrase per page, but it is not uncommon for a page to rank for a group of related long-tail queries. However if a page is an F for a query, you may have to make a lot of changes for it to grade well for that query, which would likely have a bad effect on the query that is is meant to rank for.
It would probably be better for you to figure out what elements of the page caused it to come up for the phrase it got an F for, and make a new page keeping those related elements and adding content relevant to this phrase. [The letter grades are not supposed to be the final word on the probable success of a page, btw.]
As far as whether your client is causing significant changes in Google's behavior, I would think not or else a lot of SEOs would be spending a lot of time typing in strategic search terms. [Though perhaps if they are very low search volume terms and bringing up results on the fifth page, it might not be impossible that that is what Moz is picking up. But it still wouldn't be important.]