Mank was apologizing because his script for Citizen Kane included a character that was based on Davies but was depicted as a terribly untalented performer, while Davies was quite talented in real life.
Davies was apologizing because her lover and benefactor, William Randolph Hearst, was trying to prevent Citizen Kane from being released.
From the Wikipedia entries for Citizen Kane and Marion Davies:
Many assumed that the character of Susan Alexander Kane was based on
Marion Davies, Hearst's mistress whose career he managed and whom
Hearst promoted as a motion picture actress. This assumption was a
major reason Hearst tried to destroy Citizen Kane...
By the time of her death, her popular association with the character
of Susan Alexander Kane in the film Citizen Kane (1941) already
overshadowed Davies' legacy as a talented actress. The title
character's second wife—an untalented singer whom he tries to
promote—was widely assumed to be based upon Davies. However, many
commentators, including writer-director Orson Welles, defended
Davies's record as a gifted actress and comedienne to whom Hearst's
patronage did more harm than good.