The former actress says the use of the letter was a breach of copyright, infringement of her privacy, and breaches the UK Data Protection Act.
The 'Suits' actor said part of what the pair hope to accomplish through their advocacy is to emphasise that "we all know what it feels like to have our feelings hurt."
Trump was asked about the video during a White House press conference, with the reporter saying the couple 'essentially encouraged people to vote for Joe Biden.'
Six months after detangling their work lives from the British royal family, Prince Harry and his wife, Meghan, have signed a multiyear deal to produce nature series, documentaries.
Markle and husband Prince Harry relocated to the United States via Canada this year after announcing in January that they were quitting frontline British royal duties.
Lawyers for Markle, who turned 39 on Tuesday, had argued for the friends who gave an interview to People' magazine to remain anonymous in reports of the proceedings.
The target of her lawsuit, Associated Newspapers Ltd, argues that the principle of open justice "The public's right to know" means the friends should be identified.
Royal rift: Prince William's 'offensive' relationship advice to Harry on Meghan laid bare in new book
The second installment of a serialized version of the book 'Finding Freedom,' claimed that Harry was angered by what he perceived as William's snobby tone about Meghan Markle.
The book's authors, Omid Scobie and Carolyn Durand, write about a culture of increasing tension between the Duke and Duchess of Sussex and other members of the royal family in the lead up.
The royal couple gives a detailed account in the lawsuit of the media harassment they say they've had to constantly fight in the months since they moved to Southern California.
The "institution" of the monarchy failed to protect her from the accusations and she was "prohibited from defending herself", they added.
The video of Markle, 38, reading Archie his favourite book, "Duck! Rabbit!", is recorded by Harry, 35, and has an appeal at the end for donations for Save the Children charity.
The court threw out some of the causes of action argued in her lawsuit, including the claim that the newspaper publisher acted 'dishonestly' by quoting only certain passages of her letter.
The former American actress was seeking damages from Associated Newspapers Ltd for alleged misuse of private information, copyright infringement and breach of the Data Protection Act.
The couple, who have formally stepped down as senior members of the British Royal family, said there would be 'no corroboration and zero engagement' with the newspapers,