In my local repo I have one commit with an incorrect commit message.
I've already published the incorrect commit message with
git commit --amend
Easiest solution (but please read this whole answer before doing this):
git rebase -i <hash-of-commit-preceding-the-incorrect-one>
rewordon the line for the incorrect commit.
git push --forceto update GitHub.
This will mean you will be publishing a modified version of a previously published repository. If anyone pulled or fetched from your repo between when you made the mistake with the incorrect commit message, and when you fixed it, then they will experience some difficulties later. So be sure you can accept this consequence before trying this.