could any of the kingdoms at middle earth attack the southern continents?

[Zigûr]

Are we talking about Harad here? Which, incidentally, is actually still part of "Middle-earth." I assumed originally that this thread was specifically talking about the "Dark Land", which is to say the far Southern continent which appears on one of the 1930s Ambarkanta maps of Arda and might potentially be considered a quasi-Australian or Antarctican landmass. This is to say, a region about which we know literally nothing apart from a shape and two names, "Dark Land" and "South Land", the former of which can easily be confused with Mordor and the latter with Harad.

news would have spreaded from the haradrim and corsairs

Given that "Harad" is just a catch-all name for almost everything South of Gondor, aren't all the inhabitants of the "Southern Continent" technically Haradrim? If we're not talking about the "Dark Land," isn't the "Southern Continent" just Harad?
It's been stated that there was probably conflict between the Easterlings and the Haradrim and we know for a fact that Gondor and Rohan were forced to attack certain parts of Harad (and Rhûn) in the Fourth Age to defeat realms which refused to put aside their ancient enmity with the West even after the fall of Sauron. So if by "Middle-earth" we mean "parts of Middle-earth north of Harad" then yes they did, which must as such prove that yes, they could.