The Duchess of Sussex wants the Mail On Sunday to hand over any copies of her letter to her father - and publish a front-page statement and put a notice on MailOnline for at least six months.
Meghan won her privacy case last month after a judge said the paper had breached her privacy and copyright by publishing parts of the letter.
Her lawyers are also seeking £1.5m from the Mail On Sunday to cover their legal fees, with half to be paid within two weeks.
The new details emerged today in a hearing over costs and other unresolved matters.
The duchess's lawyers want the Mail to publish statements to say she has won her case, with an online notice staying on the MailOnline homepage for "not less than 6 months".
They said they were not looking to punish the paper and were only seeking nominal damages.
Ian Mill QC, representing Meghan, also applied for an injunction, saying it was needed to stop the publication from continuing to breach her privacy and copyright.
"The defendant has still not removed the infringing articles from MailOnline," said Mr Mill in his written submission.
Her legal team also asked the High Court to order the paper to destroy any electronic copies of the letter, or any notes made about it.
The Mail wants to appeal the original ruling in Meghan's favour, and says other issues - such as whether she had sole ownership of copyright to the letter - need to be looked at.
Anthony White QC, representing the Mail's owner, said an order on handing over copies of the letter should be put on hold until any appeal is heard.
He also said publishing a statement saying the duchess had won the case appeared to be "a species of punishment or retribution" because her victory had already received much coverage in the media.
Meghan sent the five-page handwritten letter to her estranged father, Thomas Markle, in August 2018 and her lawyers had argued it was never meant to be shared by him.
But in a series of articles in February 2019 the Mail On Sunday published parts of it.
In a statement after last month's ruling, the duchess said she was grateful that Associated Newspapers and the Mail on Sunday had been "held to account for their illegal and dehumanising practices".
"These tactics (and those of their sister publications MailOnline and the Daily Mail) are not new; in fact, they've been going on for far too long without consequence," she said.
"For these outlets, it's a game. For me and so many others, it's real life, real relationships, and very real sadness. The damage they have done and continue to do runs deep.
"The world needs reliable, fact-checked, high-quality news. What The Mail on Sunday and its partner publications do is the opposite.
"We all lose when misinformation sells more than truth, when moral exploitation sells more than decency, and when companies create their business model to profit from people's pain."
Meghan and Prince Harry's much-anticipated interview with Oprah Winfrey will air on 7 March.
In a trailer released this week, Harry said he feared "history repeating itself", in reference to his mother's death, and said the process of leaving royal life had been "unbelievably tough" for him and Meghan.
The couple also recently confirmed they had decided to permanently step back from royal duties as they continue their new life in America.
