There is a newspaper report of 19.7.1729 announcing the death of Henry Moor-"eldest son of the Hon Dr Moor by Lady Rooke"....maybe George Jnr was brought up in this family?
That would have been the obvious home for him, in my opinion. That's why I asked in a previous post if George had a stepmother.
Possible alternatives:
Living with one of the guardians.
The guardians arranging for someone to look after him.
Boarding school, perhaps staying with stepmother or one of the guardians, or another person for holidays. Was there a school for sons of naval officers? (Prince Phillip didn't have a home, he stayed with schoolfriends for holidays.)
May have been a combination of any of the above depending on George's age and circumstances.
His guardians would have made decisions about his care and maintenance, perhaps in consultation with his stepmother & her husband, taking into account wishes expressed by Sir George when he was alive.
If there was any further written mention of George's childhood it was likely to have been in letters or journals /diaries. There may have been nothing written down, everything may have been arranged by verbal agreement.
Did young George have an occupation?
My most socially exalted ancestors were yeoman farmers and business-owners. Wills of any who died leaving dependent children made the main heir (eldest son or daughter) if over 21, or the widow and/or the executors responsible for care of the children. Instructions were general and basic: sufficient food & drink, washing, education, a younger son to be apprenticed when old enough. Specific decisions about a child's future (e.g. a trade or skill chosen for a son) were left to the good sense of the widow/executor/elder son/daughter who were trusted to act in the child's best interests. They lived in a small, long-established community with close ties between certain families.