People have lost their lives and others have lost everything, all their possessions, their home and everything.
"In recent months, the country has witnessed a succession of bad tragedies", she said.
May has announced a fast-tracked public inquiry into the fire and said every household affected by the blaze would receive a minimum payment of £5,500 pounds from a £5 million emergency fund.
Public anger is mounting as residents and neighbors demand answers for how the blaze early Wednesday spread so quickly and trapped so numerous tower's 600-odd residents.
"My understanding is the cladding in question, this flammable cladding which is banned in Europe and the U.S., is also banned here", Chancellor Philip Hammond said in a BBC interview.
"Government is making money available, we're ensuring that we're going to get to the bottom of what's happened, we will ensure that people are rehoused. The Council is committed to looking after the immediate and longer-term housing needs of all those affected by the fire", it said.
Cundy also said the London Fire Brigade searched "every floor of the building" and recovered 16 bodies, but added it may take weeks for the search and recovery operation to become "significant". Of those, 10 remain in critical condition.
Scores of people attended a service Sunday morning at the nearby Notting Hill Methodist Church, at which tribute was paid to those impacted by the fire.
58 missing and presumed dead in London apartment fire
He says it may be necessary for numerous outmoded tower blocks built in the 1970s to be demolished because of safety concerns.
"It has been decided today that the public inquiry will report back to me personally", May said. Numerous displaced are living in churches and community centres.
The government has said the country will observe a moment of silence at 11 a.m. Monday in remembrance of the victims. Grief turned to outrage Friday amid reports that materials used in the building's renovation could have fueled the i. No sprinkler system was installed.
May made the announcement while speaking to survivors of the fire that on Wednesday ripped through the 24-story Grenfell Tower in Kensington while most residents were sleeping, leaving dozens dead, of which 30 have been found and identified, and hundreds homeless.
"Lack of support and relief for Grenfell Tower survivors is an outrage & a scandal".
Residents who survived said they warned the building's manager about fire hazards for years but were ignored.
Lammy, who lost a friend in the fire - talented young artist Khadija Saye - also highlighted people's concerns that evidence concerning the tower's recent refurbishment might be destroyed or lost before it can be seized by police.
The British Prime Minister met a group of victims, residents, volunteers and community leaders at a church close to the scene of the horrific blaze, and earlier visited survivors in hospital. Britain's Press Association says around 70 people are missing.
May also said neighborhood councils had been ordered to complete safety checks on high-rise buildings in their area as a matter of urgency.
May herself had come in for a barrage of criticism for failing to visit residents and her response to the disaster.