Short Answer

Normally, you should only ask Allah for help. There is no need to think that Allah will not answer your prayers due to your sins so you can get around that by having a righteous person ask Allah on your behalf. Nonetheless, it is fine to ask other people to pray for you, since the Prophet even asked his Companions to pray for him. However, you should never ask a dead person to pray for you such as saying, “O Prophet, please ask Allah to forgive my sins”. Also, it is allowed to mention the Prophet or a righteous person when asking Allah for something, since this action was approved by the Prophet a few times. Mentioning the Prophet or a righteous person in your supplication is like saying, “Allah, I have so few good deeds, but I love people You love, like the Prophet. So count that as a good deed for me and accept my request.”

What the Qur’an Says

“You alone do we worship, and from You alone do we seek help.” (Qur’an 1: 5)

What the Sunnah Says

When there was a severe drought, Umar ibn al-Khattab asked Al-Abbas ibn Abdul Muttalib to supplicate for Allah to send rain. Umar would say, “O Allah, we used to ask our Prophet, peace and blessings be upon him, to supplicate for rain and you would bless us with rain. Now we ask the uncle of our Prophet to supplicate for rain, so bless us with rain.” And so it would rain. (Ṣaḥīḥ al-Bukhārī #1010)

The Prophet told Uthman ibn Hunayf to make this supplication: ‘O Allah, I ask You and turn towards You by Your Prophet Muhammad, the Prophet of Mercy. Indeed, I have turned to my Lord, by means of You, concerning this need of mine, so that it can be resolved, so O Allah so accept his intercession for me.” Tirmidhi #3578 (he graded it hasan sahih and Albani graded it sahih)

`Umar sought permission from the Prophet concerning `Umrah, so he said: “O my little brother, include us in your supplication and do not forget us.” (Tirmidhi #3562 hasan sahih)

What the Scholars Said

Imam Ibn Taymiyyah said, “We know that Umar and the senior companions did not consider this to be legislated after the death of the Prophet as it was during his life. Rather, they used to ask him to supplicate for rain and use him as a means, but after he died they no longer used him as a means.” (Qāʻidah Jalīlah fī al-Tawassul wal-Wasīlah 1:228) He also said, “When they replaced using the Prophet as a means for using Al-Abbas instead, it was understood that what they did in his life became impossible after his death, unlike using him as a means by having faith in him and obeying him, which are legislated for all times.” (Majmū’ al-Fatāwá 1:201)

Advice

Remember that asking another person to pray for you or mentioning a righteous person is not required or even recommended. It will not help your prayer get accepted. Most of the Companions never did this at all.