BBC director general Tony Hall has apologised and said a mistake was made after a news report containing a racial slur was broadcast last month.
The N-word was used in full in a report about a racially aggravated attack in Bristol, broadcast by Points West and the BBC News Channel on 29 July.
Lord Hall said he now accepts the BBC should have taken a different approach.
On Saturday, BBC Radio 1Xtra DJ Sideman – real name David Whitely – quit the station over the BBC’s use of the N-word.
He said “the action and the defence of the action feels like a slap in the face of our community”.
His actions were backed by a number of politicians and BBC staff, who offered support to the DJ.
In an email, sent to all BBC staff, Lord Hall said: “I recognise that we have ended up creating distress amongst many people.”
His statement followed high-level discussions with BBC colleagues on Sunday morning.
The Points West story described an attack on a 21-year-old NHS worker and musician known as K or K-Dogg, who was hit by a car on 22 July while walking to a bus stop from his workplace, Southmead Hospital in Bristol.
In his message, Lord Hall emphasised it was “the BBC’s intention was to highlight an alleged racist attack”.
“This is important journalism which the BBC should be reporting on and we will continue to do so,” he said. “Yet despite these good intentions, I recognise that we have ended up creating distress amongst many people.
“The BBC now accepts that we should have taken a different approach at the time of broadcast and we are very sorry for that. We will now be strengthening our guidance on offensive language across our output.
“Every organisation should be able to acknowledge when it has made a mistake. We made one here.”
K-Dogg suffered serious injuries including a broken leg, nose and cheekbone in the attack.
Police said the incident is being treated as racially aggravated due to the racist language used by the occupants of the car.
In its initial defence, the BBC said that the organisation felt it needed “to explain, and report, not just the injuries but, given their alleged extreme nature, the words alleged to have been used”.
It said at the time that the decision, which was supported by K-Dogg’s family, had not been taken lightly and that the BBC understood people would be upset.