To pass over lightly or without due mention or consideration (often followed by over). (Dictionary.com)
To pronounce (a syllable, word, etc.) indistinctly by combining, reducing, or omitting sounds, as in hurried or careless utterance. (Dictionary.com)
Indistinct and difficult to hear or understand. (Collins Dictionary)
To pronounce the sounds of a word in a way that is wrong or not clear. (Cambridge Dictionary)
Speak (words) indistinctly so that the sounds run into one another. (Oxford Dictionaries)
Pass over (a fact or aspect) so as to conceal or minimise it. (Oxford Dictionaries)
A slur is an insulting remark which could damage someone’s reputation. (Collins Dictionary)
If someone slurs their speech or if their speech slurs, they do not pronounce each word clearly, because they are drunk, ill, or sleepy. (Collins Dictionary)
Examples
Brookwood High band members hit jarring note with racial slur.
https://www.myajc.com/blog/get-schooled/brookwood-high-band-members-hit-jarring-note-with-racial-slur/BqVNqfllHb3ySKcqm8R1DM/
Wake sheriff bars deputies from Fuquay bagel shop over racial slur.
https://www.wral.com/wake-sheriff-bars-deputies-from-fuquay-bagel-shop-over-racial-slur/18047526/
Kansas recalls 731 license plates over ethnic slur.
https://www.autoblog.com/2018/11/28/kansas-recalls-license-plates-racial-slur/?guccounter=1