Paki is certainly considered a racial slur in the UK.
In Harry’s case, he clearly didn’t show any animosity and clearly thought of the man as his friend and I don’t think he was anymore racist than the average person, however that doesn’t mean that it’s not a racial slur anymore than the fact that some white people used to talk affectionately about “our niggers” meant that nigger isn’t a racial slur.
Similarly, Jesse Jackson most certainly wasn’t an anti-Semite, but that didn’t mean that referring to New York City as “hymietown” wasn’t a racial(or ethnic for the pedants) slur.
Also, I don’t think Prince Harry had to go to any racial/diversity awareness type course(that’s more an American thing), though he was forced to apologize and it was clearly embarrassing for him.