A statement from Buckingham Palace states that even if King Charles is unwell and not able to carry out his duties or if he is away from home, Prince Harry will not be called to fill in for him. The same is applicable to Prince Andrew as well.
The list of people who could be asked to serve as Councillor of State now includes Prince Edward, Earl of Wessex and Princess Anne. Prince Harry and his wife Meghan had stepped down from royal duties and moved to California in January 2020.
Prince Andrew quit his public position in May 2020 due to criticism over his friendship with convicted sex offender Jeffrey Epstein. The Queen had also terminated his honorary royal charitable patronage and military affiliations in January 2022.