The 1917 LPs of George V give the style of HRH Prince or Princess to:
The children of the monarch - Charles, Anne, Andrew and Edward
The male line grandchildren of the monarch - William, Harry, Beatrice, Eugenie, Louise, James (last two is debateable due to othe announcements and I am not going to debate it here), Richard, Edward, Alexandra and Michael.
The eldest son of the eldest son of the Prince of Wales - George (although QEII has amended this last clause to be ALL the childen of Prince William)
As things currently stand only William's children will be born as Prince/Princess in the current reign as Harry's children aren't covered for that styling under any existing LPs. Harry's children will be HRH once Charles becomes King, unless new LPs are issued to restrict it or he follows Edward's lead and says that they won't be so labelled.
Prince Michael's children, like Prince Richard (Duke of Gloucester) and Prince Edward (Duke of Kent) will not be HRH Prince or Princess at all of these children will be great-grandchildren of a monarch but not through the primary line.
Prince Michael, being the younger son of a son of a monarch, has children who are given the styling of the children of a Duke - again under the 1917 LPs. Fred's children will be simply Mr and Miss. This is NOT the normal situation for the children of the younger son of a Duke but special for the younger sons of royal Dukes.
Whereas the children of a princess go to Mr or Miss as grandchildren of the monarch but from princes it takes at least to great-grandchildren before that happens - further example of the sexism in the BRF.
The idea with the 1917 LPs are that there will always be a limited number of people who had HRH Prince/Princess.
Currently there are - 15 by birth, including Louise and James with only William and Harry in the younger generation able to have children with that styling and Harry only after his grandmother dies and his father becomes King. If Charles doesn't outlive his mother then Harry's children won't be Prince/Princess as they won't be the male line grandchildren of a monarch.