Does the saying "The prayers are expiations for what occurs between them, so long as major sins are avoided" apply to one who committed a major sin in his youth and then repented from it? And does his not having avoided major sins in his youth prevent the prayers from atoning for him?

Do not worry or grieve, for Allah the Almighty is All-Merciful, All-Forgiving. As long as you have repented to Allah from this grave sin, Allah the Almighty will forgive it. Allah the Almighty says: "But indeed, I am the All-Forgiving to whoever repents and believes and does righteousness and then is guided." [Taha: 82]. And Allah the Almighty says: "Say, "O My servants who have transgressed against themselves [by sinning], do not despair of the mercy of Allah. Indeed, Allah forgives all sins. Indeed, it is He who is the All-Forgiving, the All-Merciful." [Az-Zumar: 53].

Increase in good deeds and supplicate to Allah to accept your repentance. Beware of despair and hopelessness in Allah's mercy and forgiveness, for Satan tempts a person to cause him to fall into sin, then tries to make him despair of the acceptance of his repentance. Allah the Almighty says: "And who despairs of the mercy of his Lord except for those astray?" [Al-Hijr: 56]. And Allah the Almighty says: "And do not despair of the relief from Allah. Indeed, no one despairs of relief from Allah except the disbelieving people." [Yusuf: 87].

Rejoice in Allah's acceptance of your repentance and His satisfaction with you, for the one who repents from sin is beloved by Allah. Allah the Almighty says: "Indeed, Allah loves those who repent and loves those who purify themselves." [Al-Baqarah: 222].