Correction Note or New Paper?
That depends on one question: If we assume that the original paper would not contain this error, would your new related ideas be enough for a publication?

If yes then go ahead and publish it, including the correction (and of course notify the journal about it). If needed put a bit more research into the topic to get enough new results.

If no then this might seem like an excuse to get a new publication with content that is not worthy of publication (yet), so it might look like you try to cheat on your paper count, which is of course bad for your professional reputation.