NOTE. Kaikeyī-devī has already [at least superficially] begged forgiveness for her ill deed and yet Lord Rāmacandra is mindful of her ambitions because external circumstances may forcibly prevent our material desires from being fulfilled, but that doesn’t mean that those material desires have disappeared from our minds.

Those material desires can be removed if one firmly engages in devotional service to the Supreme Personality of Godhead in bhakti-yoga. Even other processes such as jñāna-yoga or aṣṭāṅga-yoga cannot help us succeed in this matter, as noted by Prabhupāda in his purport to Bhagavad-gītā 2.59:

Unless one is transcendentally situated, it is not possible to cease from sense enjoyment. The process of restriction from sense enjoyment by rules and regulations is something like restricting a diseased person from certain types of eatables. The patient, however, neither likes such restrictions nor loses his taste for eatables. Similarly, sense restriction by some spiritual process like aṣṭāṅga-yoga, in the matter of yama, niyama, āsana, prāṇāyāma, pratyāhāra, dhāraṇā, dhyāna, etc., is recommended for less intelligent persons who have no better knowledge. But one who has tasted the beauty of the Supreme Lord Kṛṣṇa, in the course of his advancement in Kṛṣṇa consciousness, no longer has a taste for dead, material things. Therefore, restrictions are there for the less intelligent neophytes in the spiritual advancement of life, but such restrictions are only good until one actually has a taste for Kṛṣṇa consciousness. When one is actually Kṛṣṇa conscious, he automatically loses his taste for pale things.

Śrīmad-Bhāgavatam 4.22.39 confirms this as follows:

yat-pāda-paṅkaja-palāśa-vilāsa-bhaktyā
karmāśayaṁ grathitam udgrathayanti santaḥ
tadvan na rikta-matayo yatayo ’pi ruddha-
sroto-gaṇās tam araṇaṁ bhaja vāsudevam

The devotees, who are always engaged in the service of the toes of the lotus feet of the Lord, can very easily overcome hard-knotted desires for fruitive activities. Because this is very difficult, the nondevotees—the jñānīs and yogīs—although trying to stop the waves of sense gratification, cannot do so. Therefore you are advised to engage in the devotional service of Kṛṣṇa, the son of Vasudeva.