Does regret help us to correct our faults?

Suppose a person was working under the sun for a hard days toil. He grew thirsty and went inside to find something to quench his thirst. Upon the table he saw a clear glass of liquid and assuming it was water that his wife prepared for him, he rushed and carelessly gulped down the whole thing. From the odd taste he quickly suspected something was wrong. Turning the glass around in his hand he noticed a poison symbol. The glass was previously full of a household cleaning solution. Instantly, the man was filled with immense regret of a very profound sort.

Rushing to the hospital his mind would not be consumed with an ounce of guilt, but rather would be full of regret. He would survive this experience and the regret he felt would act as a protection against committing the same mistake in the future. From then on, his regret acted as a cause for him to always check the label before gulping down a glass of liquid.

This is the difference between regret and guilt and the above story is how regret should be understood and disambiguated from guilt. Regret for committing non-virtuous acts or for failing to commit virtuous acts can act as the protector of repeating the same mistake in the future. On the other hand, guilt is a useless non-virtuous emotion tending to revel in self-loathing with no positive purpose.

It is very important to understand the difference between regret and guilt. Think of the poison analogy whenever you have feelings where you are confused as to whether regret or guilt is being felt and you are trying to discern which is which.

Hope this helps!

In the Abhidhamma, regret and remorse over the past is considered an unwholesome mental factor.

Worry (kukkucca) is remorse, brooding, and repenting over evil acts done in the past or good acts left undone.

The following theme for reflection from AN 5.57 (quoted below) can prevent the arising of unarisen unwholesome states. It encourages shame of evil (hiri) and fear of evil (otappa), which are mentioned as beautiful mental factors in the Abhidhamma.

So, remorse and regret over the past is unwholesome. But shame of evil and fear of evil, or basically moral shame (hiri) and fear of wrongdoing (otappa), for the future, is wholesome.

“And for the sake of what benefit should a woman or a man, a householder or one gone forth, often reflect thus: ‘I am the owner of my kamma, the heir of my kamma; I have kamma as my origin, kamma as my relative, kamma as my resort; I will be the heir of whatever kamma, good or bad, that I do’? People engage in misconduct by body, speech, and mind. But when one often reflects upon this theme, such misconduct is either completely abandoned or diminished. It is for the sake of this benefit that a woman or a man, a householder or one gone forth, should often reflect thus: ‘I am the owner of my kamma, the heir of my kamma; I have kamma as my origin, kamma as my relative, kamma as my resort; I will be the heir of whatever kamma, good or bad, that I do.’

Hiri and otappa are explained below in Ven. Bodhi's "The Guardians of the World" :

The Buddha points to two mental qualities as the underlying safeguards of morality, thus as the protectors of both the individual and society as a whole. These two qualities are called in Pali hiri and ottappa. Hiri is an innate sense of shame over moral transgression; ottappa is moral dread, fear of the results of wrongdoing. The Buddha calls these two states the bright guardians of the world (sukka lokapala). He gives them this designation because as long as these two states prevail in people's hearts the moral standards of the world remain intact, while when their influence wanes the human world falls into unabashed promiscuity and violence, becoming almost indistinguishable from the animal realm (Itiv. 42).

While moral shame and fear of wrongdoing are united in the common task of protecting the mind from moral defilement, they differ in their individual characteristics and modes of operation. Hiri, the sense of shame, has an internal reference; it is rooted in self-respect and induces us to shrink from wrongdoing out of a feeling of personal honor. Ottappa, fear of wrongdoing, has an external orientation. It is the voice of conscience that warns us of the dire consequences of moral transgression: blame and punishment by others, the painful kammic results of evil deeds, the impediment to our desire for liberation from suffering. Acariya Buddhaghosa illustrates the difference between the two with the simile of an iron rod smeared with excrement at one end and heated to a glow at the other end: hiri is like one's disgust at grabbing the rod in the place where it is smeared with excrement, ottappa is like one's fear of grabbing it in the place where it is red hot.

And from Iti 42:

"Bhikkhus, these two bright principles protect the world. What are the two? Shame and fear of wrongdoing. If, bhikkhus, these two bright principles did not protect the world, there would not be discerned respect for mother or maternal aunt or maternal uncle's wife or a teacher's wife or the wives of other honored persons, and the world would have fallen into promiscuity, as with goats, sheep, chickens, pigs, dogs, and jackals. But as these two bright principles protect the world, there is discerned respect for mother... and the wives of other honored persons."

Those in whom shame and fear of wrong
Are not consistently found
Have deviated from the bright root
And are led back to birth and death.

But those in whom shame and fear of wrong
Are consistently ever present,
Peaceful, mature in the holy life,
They put an end to renewal of being.

Isnt the pain from our faults push us to the correct path?

Pain and suffering certainly pushes people towards religion and spirituality in search for answers and a possible solution. At the same time pain and suffering, if used correctly as vehicles, can become fertilizer for spiritual growth.

What pushes us further on the (correct) path is clarity of mind, to see clearly, also known as Right View (Sammā-Diṭṭhi).