Moderators can add "post notices" as a banner at the top of the post; there are some custom notices for certain sites, three moderator-added notices that exist network-wide, and additional system notices for bounties and protected/locked questions. More info at:
The "needs citation" notice added by moderators appears to users as:
Want to improve this post? Add citations from reputable sources by editing the post (link to edit). Posts with unsourced content may be edited or deleted.
and originated on Skeptics.SE. It is also used on other sites like Biology.SE to mark answers that fall short of requirements for suitable references on those sites.
Usually the notices are added when a moderator sees an unreferenced answer or in response to a custom flag alerting moderators to add the banner. The notice is clear about what a user (OP or otherwise) should do: edit the post to add references. However, it is not clear about what should happen next. Compared to the other two network-wide notices, this one is most temporary and should be removed after a sufficient edit (presumably "current event" is also temporary, but the timeline of what makes an event "current" isn't particularly related to any user action, only the inexorable transition of present into past).
It seems up to site moderator teams to monitor the posts with these banners and act on them accordingly (removing posts that remain unreferenced; removing the banner for posts edited adequately); that's something I personally could certainly be better about, as most posts that I add this banner to I may never check on again. We do have a tool to let us see all the bannered posts in one place, which is very helpful. However, even in an ideal world, the wait is a bit frustrating to the user, who can do nothing in the meantime:
The action a user should take is to raise a custom flag (Flag/In need of moderator intervention), but it seems the only way they would be informed of this would be if the moderator left a separate comment every time they banner the post. These notices apply disproportionately to newer users because they're about a site guideline that veterans are typically aware of, so I don't think it's reasonable to expect them to be familiar with flags. I'll note that one of the meta questions was raised by a user with 20,000 rep (and, frankly, it takes a bit of site familiarity to realize that meta is even a place to go when wondering about something like this).
My understanding is that Biology.SE or some other site could request a custom post notice similar to this one that includes further instruction, but I think a simpler solution that would also retroactively label any currently noticed posts would be to add another instruction to the notice, such as:
Want to improve this post? Add citations from reputable sources by editing the post (link to edit). Posts with unsourced content may be edited or deleted. Flag "in need of moderator intervention" once the post has been edited with citations to have this notice removed.