Obviously, The Prince and the Showgirl, referred to as The Sleeping Prince in the film did happen. However, whether the actual love scenes between Colin Clark and Marilyn actually happened is highly doubtful. The Prince, the Showgirl and Me was Colin’s first book which didn’t mention the weeklong affair.
Marilyn had an entourage for the majority of her time in England. You can read about Marilyn's life in England in detail in Michelle Morgan's book 'When Marilyn Met the Queen.'