The answer to this question lies more within the realm of the puranas and the words of saints rather than in sruti. The great saint Swami Ramdas (https://en.wikipedia.org/wiki/Swami_Ramdas) was a great devotee of Sri Rama and chanting Ram Nam. Swami Ramdas said (Thus Speaks Ramdas):
All names of God are great and powerful. But Ram Nam has a value all its own. Rhythm is a most important factor in a Mantra. The soothing melody of the repeated word, or combination of words, has a marvelous effect on the distracted mind. No two letters linked together could, by their harmonious music, lull and bring peace to the mind more than Ram. It is said, rightly, that 'Ram' is equal in power to the sacred symbol 'OM'.
Sri Samarth Ramdas assures the aspirant that if he takes "Sri Ram Jai Ram Jai Jai Ram" 13 crores of times, he will have the vision of Sir Rama. The Mantra mentioned by Samarth Ramdas is without 'OM'. The Mantra Ramdas gives you is "OM Sri Ram Jai Ram Jai Jai Ram". 'OM' has untold spiritual power. Hence Ramdas, from his own experience tells you that by repeating this Mantra six crores of times, you will attain salvation. Repeat the Mantra at all times, until the target is reached. You need not keep count of the Mantra. When it reached six crores you will automatically realize Ram--the Supreme Self.
and in the Srimad Devi Puranam (Chapter XXVI):
6-14. Nârâyana said :-- O Muni! The King As’vapati reigned in Bhadrades’a, rendering his enemies powerless and making his friends painless. He had a queen very religious; her name was Mâlatî; She was like a second Laksmî. She was barren; and desirous of an issue, She under the instruction of Vas’istha, duly worshipped Sâvitrî with devotion. But She did not receive any vision nor any command; therefore She returned home with a grievous heart. Seeing her sorry, the king consoled her with good words and himself accompanied her to
Puskara with a view to perform Tapas to Sâvitrî with devotion and, being self-controlled, practised tapasyâ for one hundred years. Still he could not see Sâvitrî, but voice came to him. An incorporeal, celestial voice reached his ears :-- “Perform Japam (repeat) ten lakhs of Gâyatrî Mantram.” At this moment Parâsara came up there. The king bowed down to him. The Muni said :-- O King! One japa of Gâyatrî, destroys the days sins. Ten Japams of Gâyatrî destroy day and night’s sins.
15-40. One hundred Gâyatrî Japams destroy one month’s sins. One thousand Japams destroy one year’s sins. One lakh Gâyatrî Japams destroy the sins of the present birth and ten lakh Gâyatrî Japams destroy the sins of other births. One hundred lakhs of Japams destroy the sins of all the births. If ten times that (i.e., 1,000 lakhs) be done, then liberation is obtained.
And as stated in my comment above, Sarada Devi, the wife of Ramakrishna Paramahamsa said:
One has to suffer the consequences of one's deeds. But by repeating the name of God, you can lessen its intensity. If you were destined to have a wound as wide as a ploughshare, you will get a pinprick at least. The effect of karma can be counteracted to a great extent by Japa and austerities.
The doing of charity, while it can create good karma, does not of itself dilute the effects of prarabdha karma.
Sankara addresses the effects of prarabdha karma in his Aparoksanubhuti (Swami Vimuktananda translator):
- The theory one hears from the scripture that Prarabdha does not hold upon one even after the origination of the knowledge of Atman, is now being refuted.
From the scripture--From such scriptural texts as: "The delay in his case os only so long as he is not released (from the body), then he will attain Brahman" (Chhandogya Upanishad 6.14.2).
Prarabdha does not lose its hold, etc.--The Shruti in many places has declared that even a Jnani is not free from the operation of Prarabdha. Shankara has dealt with this point at length in his commentaries on Chhandogya Upanishad (6.14.2), Vedanta Sutras (4.1.15), and Gita (4.37). In all those places he has supported the popular view that Prarabdha is binding on even on the Jnani. But here as well as in his Vivekachudamani (453-463) he has boldly asserted the true Vedantic view without any compromise. He has clearly shown that to a Jnani there is no such thing as a body, and it is meaningless to say that he is any longer under the influence of Prarabdha, which has no hold upon the bodiless Atman. The author brings his arguments in support if this view in stanzas 91 and 92.
- After the origination of the knowledge of Reality Prarabdha verily ceases to exist, insomuch as the body and the like becomes non-existent; just as a dream does not exist on waking.
[translator's commentary] Insomuch as the body etc.--The body, mind, intelligence and the like have their existence only in ignorance and therefore cannot exist when the latter is entirely destroyed by Knowledge. In the absence of the body, Prarabdha also necessarily ceases to exist, since there remains nothing on which it can act.
And Brahma Sutras (3.2.38-41) says that it is from Isvara alone and not one's karma that the fruit of one's actions comes from. Sankara in his commentary to verse 41 quotes the Gita (7.21-22):
Whichever divine form a devotee wishes to worship...and obtains from it the results he desires, as ordained by Me.
Karma, including Prarabdha karma, is ordained by the Lord and can be lessened or completely negated by the Lord. It is not automatic.