I found some more info on Princess Marie in Roger Fulford's " Dearest Mama".
In the lead up to Princess Alexandra of Denmark's marriage to the Prince of Wales, Queen Victoria had heard that Princess Louise (Alexandra's mother) had had an illegitmate baby. Apparently the Duchess of Nassau (Princess Adelheid of Anhalt, daughter of Princess Marie of Hesse) had been spreading the story, she was a neice of Princess Louise.
Queen Victoria asked her eldest daughter to do some investigating, and on January 13, 1862 Vicky reported to her mother the following:-
"The mother of the Duchess, Princess C.'s sister is bad and has had an illegitimate child; now either she has told her daughter this was Princess C.'s or, the Duchess of N. has heard the report about her mother, puts it upon Princess C., her aunt, to screen her mother, which, though inexcusable to me, seems far from unlikely."
I have never heard any thing about this illegitimate child.