All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
The Hadeeth in question was narrated by Abu Daawood and by At-Tirmithi through a good chain of narrators. It has also been narrated through many other chains: there is a narration in Al-Bukhari that substantiates it; Abu Hurayrah narrated on the authority of the Prophet, sallallaahu ‘alayhi wa sallam, that every prophet had one answered prayer, and the Prophet, sallallaahu ‘alayhi wa sallam, saved his prayer to intercede for his nation on the Day of Judgment. [Al-Bukhari and Muslim]
The word “Muslim” indicates that the intercession will apply to every Muslim who has died without associating partners with Allaah The Almighty.
Therefore, the Hadeeth you are asking about is authentic, as explained above. It is also authentically established that the intercession will be for those Muslims who have committed sins. However, it must not be understood as a call for proceeding to commit minor and major sins, or belittling sins in the assurance of gaining the intercession of the Prophet, sallallaahu ‘alayhi wa sallam, and the abundant grace and mercy of Allaah The Almighty. This is because Allaah The Almighty is severe in inflicting punishment and Watchful of transgressions against His laws and limits, just as He is Merciful and Forgiving.
Just as it was established that the Prophet, sallallaahu ‘alayhi wa sallam, would intercede for some people from his nation, it was also established that some people from his nation would be prevented from drinking from his Hawdh (basin), from which if a person takes one draught, they will not get thirsty forever. [Al-Bukhari and Muslim]
Therefore, every Muslim ought not to rely solely on the promise of intercession and neglect the promises of punishment, so that he will not indulge in prohibited deeds and instead be assured of being safe from the punishment of Allaah Who Says (what means): {Then did they feel secure from the plan of Allaah? But no one feels secure from the plan of Allah except the losing people.}[Quran 7:99]
It has also been established that faith increases when good deeds are done, and decreases when there is little practice of good deeds and many sins are committed. Finally the heart is so shrouded that it will not be occupied with faith. Allaah the Almighty Says (what means): {No! Rather, the stain has covered their hearts of that which they were earning.}[Quran 83:14]
Thus, continuing to commit sins may lead to loss of faith and perhaps death in a state of disbelief. At that time, no intercession will help such a person and they will be an eternal dweller of Hell.
And Allaah Knows best.