One woman wept, saying it was because the Prime Minister had declined to speak to those waiting outside to speak to her.
Rev. Mark O'Donoghue said Mrs May had expressed what came across as a "genuine compassion" as he described her reaction as something that was "long overdue".
Ms Leadsom managed to visit victims but was heckled on her trip, with one man laying into the leader of the House of Commons for Mrs May's response.
She further explained that a public inquiry would be strongly "government-led, government-controlled, government outcome", and would exclude residents nearly entirely.
Sixteen "very ordinary people" sat in Downing Street to bring their concerns to Theresa May in an "unprecedented" meeting and finally felt they were listened to, the Bishop of Kensington, Dr Graham Tomlin said.
Eleanor Kelly, chief executive of Southwark Council, on behalf of the newly established Grenfell Fire Response Team, said: "We want to make clear that whilst the emergency and local community response was nothing short of heroic, we know that the initial response was simply not good enough on the ground".
It has shown the great spirit of the people in response to a tragedy such as this.
Bill Cosby sex assault case in hands of jury
They wanted to revisit the portion of a 2005 civil deposition where the comedian, now 79, talked about giving her "three friends". Phone records showed she made no such call in January, he said, and instead show she only called Cosby's phone numbers in NY .
She has promised a £5m support fund and more staff deployed across the area to assist the bereaved families.
At least 58 people are missing and presumed dead in the blaze that broke out in the early hours, catching many families in their sleep. "Why did you hear their anger, their pain but she initially did not?" "There was passion, there was anger, but there was good, hard, reasoned argument used by the residents".
He believes that, because of the negative publicity surrounding her first visit, "we'll be seeing her forced down there today or tomorrow" to meet members of the public.
Speaking to Sky News' Sophy Ridge on Sunday, Mr McDonnell said: "People would not be living in high-rise blocks, unsafe conditions and some of them very poor quality, if we didn't have the housing crisis and part of that housing crisis is as a result of allowing housing particularly in London for use for speculative gain rather than for housing need".
Chancellor Philip Hammond said the cladding, which has been blamed for spreading the fire, is banned in Britain.
The comments came as former chief fire officer Ronnie King told the Observer newspaper that urgent requests for meetings with ministers and action to tighten rules were stonewalled.