Secretary of State Rex Tillerson seemed focused this week on rebooting his image as a beleaguered Cabinet member on the outs with his boss and his own employees — holding a rare town hall with employees, promising foreign trips into 2018 and saying he is “learning” to enjoy his job. But then he went off script by offering another invitation for diplomatic talks with nuclear-armed North Korea, putting him at odds once again with President Trump and senior White House officials, who are increasingly exasperated with the secretary of state and say he cannot remain in his job for the long term. The episode highlights the deep distrust between the White House and Tillerson and suggests how difficult it will be for the relationship to continue. While Trump and Tillerson have clashed on several policy issues — including negotiating with North Korea, the US withdrawal from the Paris climate accord and planning to move the US Embassy to Jerusalem — much of the distance between them seems personal and probably irreversible, White House officials said. Tillerson, one White House official said, “had not learned his lesson from the last time,” when Trump publicly rebuked his top diplomat on Twitter over the wisdom of talking to North Korea. A senior US official said foreign diplomats and leaders often ask if Tillerson is speaking for the administration and when he will depart. Another White House aide said White House officials, diplomats and other Cabinet secretaries largely deem the former ExxonMobil chief executive “irrelevant.” Inside the White House, this person said, there are fairly regular conversations about who will replace Tillerson even as he remains in the job. CIA Director Mike Pompeo, for example, may no longer be the leading choice because it means he would not brief Trump every day, and the president likes him in that role, the official said. “I think our allies know at this point he’s not really speaking for the administration,” this Trump official said — a particularly sharp slap given that Tillerson has sought to be a buffer and interpreter for allies angry or bewildered by some of Trump’s actions. West Wing officials spoke about Tillerson on the condition of anonymity to describe internal personnel dynamics. Raj Shah, a White House spokesman, said Trump “is very pleased with his entire national security team, which includes Secretary Tillerson.” State Department spokeswoman Heather Nauert said Tillerson “enjoys a strong relationship with the president. Most importantly, they share a commitment to the denuclearization of the Korean Peninsula, the security of our homeland and the protection of our allies.” Nauert noted that Tillerson had lunch with Trump on Thursday. The discussion included North Korea, she said. The latest dust-up began Tuesday when Tillerson caught the White House by surprise with remarks at the Atlantic Council that appeared to mark a shift away from the Trump administration’s demand that North Korea commit to disarmament upfront. Tillerson’s comments also stood in sharp contrast to Trump’s past pronouncements that talking to North Korea is a trap or waste of time.  “We’re ready to talk anytime North Korea would like to talk,” Tillerson said. “We’re ready to have the first meeting without preconditions.” “Let’s just meet and let’s — we can talk about the weather if you want,” Tillerson said to laughter. The comments upset Trump and several senior aides, and set off a cascade of emails and phone calls that ended with a terse statement from press secretary Sarah Huckabee Sanders. National Security Council aides were particularly frustrated with the remarks and complained loudly, administration officials say.  “The President’s views on North Korea have not changed. North Korea is acting in an unsafe way not only toward Japan, China, and South Korea, but the entire world,” Sanders said in the statement. “North Korea’s actions are not good for anyone and certainly not good for North Korea.” Senior Trump aides say that Tillerson knew he was contradicting Trump, according to White House advisers. The State Department said Tillerson’s invitation was not a break with the administration’s official position. Tillerson’s State Department continues to clash with the White House over personnel — picks are often scuttled or delayed, officials say — and Trump would sometimes react with exasperation when the secretary’s name is brought up, other officials said. One senior official said Trump will sometimes commend senior policy aide Stephen Miller for the time he clashed with Tillerson, or will mention disagreements Tillerson has had with other aides — and not take Tillerson’s side.  Inside the West Wing, several aides said people close to Trump essentially were counting down the days until Tillerson leaves, which they guess will be in February.  Trump has told advisers he is amazed at the negative news attention Tillerson receives and has said to at least one adviser that Tillerson probably gripes about him behind his back. But Trump does not believe Tillerson called him a “moron” as NBC News first reported, a White House official said. Tillerson has sought out Sen. Bob Corker (R-Tenn.) as an ally, frequently meeting with him for breakfast. And Corker has tried to solidify Tillerson’s standing by vouching for him to Trump and senior aides. But after Corker went on television and praised Tillerson, Trump complained to aides that Corker was defending the secretary of state.  For now, Trump has told advisers that he is not firing Tillerson.  The Washington Post