A few things people who don't watch top gear need to know, since it seems that its mostly people who don't like the show who want Jeremy to be sacked.
First, top gear is a comedy show in style. Cars are the subject, the scripts are comedic
Second, the joke is a referential joke in the context of the rhyme. The rhyme is known to have originally have a racist word and so as a joke he sings the song and mumbles the middle as a joke. Even then he wasn't satisfied with it. You see this in other comedies where a character or presenter (white, black, hispanic, etc) says the N-word but its bleeped. That is supposed to be a joke on the awkwardness of the word and situation. Usually followed by reactions from others, in this case it was the obvious uncomfortableness of Jeremy while he mumbled.
Third, Top Gear never is shy from pulling the punches and having honest and edgy humor. So, it is clear that they had no intent to be racist. They removed it with the intentions that either the script or the editing didn't convey the purpose as explained by Jeremy well enough, think of it as the comedian in a sitcom saying the N word in the story but the bleep wasn't perfect.
Lastly, all the people that want Jeremy to sacked are arguably not fans of his that may or may not wish that Top Gears time slot go to some wedding reality tv show or some sort of political roundtable show, in an attempt to be diverse in BBC scheduling. But they ignore that Top Gear has by far the most diverse fan base in the world. Its watched heavily and I do stress heavily in south east Asia, eastern and western Europe, north america, middle east, south america, Australia. No race or culture doesn't know the show, as well even though its a male oriented show its female viewer base is rather large.
The reason I wrote a lot is because I feel I have to defend this show because really its a part of my childhood. But also I don't really have to stretch too far. These types of accusations are so frequent in the world. it used to be that anyone that complained about a joke they didn't like no matter how subtle it was was just left to complain on their radical blogs, but today they have access to MPs and magazines.
I just realized my grammar sucks especially 3rd and 4th paragraph. I am not going to edit this. Deal with it, its 2014.
Here's a bunch of gifs of ME to show my emotion during this terrible process.