Once upon a time one could obtain this day a plenary indulgence by reciting the Supplication to the Madonna of Pompeii. The other day for this is the first Sunday of October.
With the changes to the concessions for indulgences, according to the Enchiridion Indulgentiarum, there is no longer any plenary indulgence for this prayer, notwithstanding anything you might see in some old book or on a website. For example, if you see something about Pope Leo XIII granting an indulgence, etc., that is null and void now.
However, the new Enchiridion says with concession #17, §3 that Marian prayers obtain a partial indulgence under the condition that the prayer is approved by competent authority and that it is recited with fervor in the state of grace (you don’t need confession and Communion within 8 days, nor must you recite the prayers for the Roman Pontiffs intentions for a partial indulgence). You can receive a partial indulgence, by maintaining this beautiful custom of the Supplication today.
For more about this, including the prayers, click HERE. I included background on Bl. Bartolo Longo, a converted Satanic priest! John Paul II beatified Bartolo Longo in 1980. Some of his writings form the basis of the Luminous Mysteries of the Rosary.