Is a person who fell into distress and problems because of a sin from which he repented with sincere repentance considered someone whose supplication is accepted by God out of necessity?

If you repent to Allah with sincere repentance that fulfills its conditions, then Allah Almighty will accept your repentance and erase the effect of the sin, and the sin will become as if it never existed, for the Prophet, peace and blessings be upon him, said: "The one who repents from sin is like one who has no sin."

As for the calamities resulting from the sin, if you are patient with them, you will have the reward of the patient ones. If you supplicate to Allah to remove the harm, it is hoped that He will respond to you, and it does not matter that the calamity stemmed from a sin you repented from. So strive in supplication, increase in good deeds, and have a good opinion of your Lord.