The paper says the memo was written after the then Foreign Secretary Boris Johnson appealed to the US in 2018 tostick with the deal.
The latest leak came despite the Met Police warning against publication.
The first memos criticising President Trump's administration, which emerged a week ago, prompted a furious reaction from the US president and resulted in Sir Kim resigning from his roles.
The Mail on Sunday reports that Sir Kim wrote to Mr Johnson informing him Republican President Trump appeared to be abandoning the nuclear deal for "personality reasons" - because the pact had been agreed by his Democrat predecessor, Barack Obama.
Under the 2015 deal backed by the US and five other nations, Iran agreed to limit its sensitive nuclear activities in return for the lifting of crippling economic sanctions.
However, President Trump said he did not think that the deal went far enough in curtailing Iran's nuclear ambitions and reinstated US sanctions after withdrawing from it in May 2018.
In atweetfrom June this year, the president also said he objected to Mr Obama having given Iran 1.8bn (1.4bn) as part of the deal. Commentators later pointed out this was related to the settlement of an unfulfilled military order from the 1970s.
The British ambassador's memo is said to have highlighted splits amongst US presidential advisers; he wrote that the White House did not have a strategy of how to proceed following withdrawal from the deal.
According to the paper, in his memo to Mr Johnson, Sir Kim wrote: "The outcome illustrated the paradox of this White House: you got exceptional access, seeing everyone short of the president; but on the substance, the administration is set upon an act of diplomatic vandalism, seemingly for ideological and personality reasons - it was Obama's deal.
"Moreover, they can't articulate any 'day-after' strategy; and contacts with State Department this morning suggest no sort of plan for reaching out to partners and allies, whether in Europe or the region."
Why did the police warn the media?
Scotland Yard's Assistant Commissioner Neil Basu warned media organisations against publishing leaked government documents, saying it "could be a criminal matter".
The warning prompted a backlash from newspaper editors and MPs on Saturday, defending the freedom of the press.
The Met Police then released a second statement making clear journalists who released further details of the former ambassador's communications could be in breach of the Official Secrets Act, in which case there is no public interest defence in law.
The Mail on Sunday said it was in the public interest to publish the memos.
The paper argued that the most recently published emails revealed "important information" on the UK's attempts to stop President Trump abandoning the Iran nuclear deal.
A spokesman said: "What could be more in the public interest than a better understanding of how this position was reached, which may have serious consequences for world peace?"
In response, a Foreign Office spokesman called it a "totally unacceptable leak" of "sensitive material" and called for the source of the leak to "face the consequences of their actions".
He added that it was "not news" that the UK and US differ in their approach to preventing Iran acquiring a nuclear weapon.
Work and Pensions Secretary Amber Rudd said she supported the paper's decision to disclose the information, adding "we have a very precious freedom of press legislation here".
What was the fallout from the initial leak?
The first leaked emails saw Sir Kim, the then UK ambassador, refer to the Trump administration as "clumsy and inept".
The US president responded by calling Sir Kim as "a very stupid guy", adding that he would no longer deal with him.