Sarah Williment, 34, said her son Jack was rushed to the Leeds hospital by ambulance with suspected pneumonia on December 3.
However, during the busiest week the A&E had seen since 2016, Sarah said he was left in a clinical treatment room for over four hours.
A photograph sent to the Yorkshire Evening Post by Sarah shows Jack with an oxygen mask, lying on a pile of coats for comfort on the floor of the room.
The boss of Leeds Teaching Hospitals NHS Trust, which runs the A&E department, personally apologised to Jack's mum.
Prime Minister Boris Johnson was questioned by broadcasters on the election campaign trail in Grimsby on Monday morning. (Dec 9)
The Prime Minister was asked about the story, first reported in the Yorkshire Evening Post on Sunday.
Asked if he had an apology for the family, he replied: "Yeah, I do, and this is exactly why we need to move on.
"We're putting £34 billion into the NHS now, this is the biggest investment we've seen in modern times but we need to drive it forward."
Mr Johnson said there is "so much more that needs to be done", adding: "That is why we just need to unblock the politics of this country and move forward."
Pressed on what he said to people who wait more than 12 hours in A&E for treatment, the PM said: "I want everybody to have the best possible experience in the NHS and I have every sympathy with people who don't."
He said he believes the NHS on the whole does a "wonderful, wonderful job", adding: "We need to be putting money in now and move forward as a country."
Speaking to LBC the Prime Minister said he wanted to apologise to "everyone who has a bad experience".
The Prime Minister added: "By and large I think the NHS do an amazing job, and I think they deserve all praise for the service they provide.
"But they do need investment and that's why we're doing it now.
"But they need investment from a One Nation government that really cares and understands - that's us that cares and understands - and you need long-term funding.
"What you cant do is have an economic disaster under Corbyn and McDonnell who have captured the far-left agenda."
Dr Yvette Oade, chief medical officer at Leeds Teaching Hospitals NHS Trust said: “Our hospitals are extremely busy at the moment and we are very sorry that Jack’s family had a long wait in our Emergency Department. Our Chief Executive Julian Hartley has spoken to Jack’s mum and offered a personal apology.
“We have seen a significant increase in the number of people visiting our Paediatric Emergency Department, and this week we saw the highest attendances we have seen since April 2016. Despite this, our staff are working tirelessly to provide the best possible care under these extreme pressures.
“Jack was quickly assessed upon arrival and seen in two different clinical treatment rooms in the Paediatric Emergency Department.
“Within four hours a decision was made to admit Jack to our Children’s Assessment and Treatment (CAT) Unit for further monitoring overnight. Unfortunately, the unit was also experiencing exceptionally high levels of demand which meant that Jack was required to wait in the clinical treatment room in the Paediatric Emergency Department until a bed became available. Jack was admitted to the CAT Unit later that evening and was discharged home the following morning after a medical review.
“We are extremely sorry that there were only chairs available in the treatment room, and no bed. This falls below our usual high standards, and for this we would like to sincerely apologise to Jack and his family.
“We are increasing the bed availability in our Children’s Hospital and our Children’s Assessment and Treatment Unit will be relocating to a larger area in the new year. We are continuing to develop the plans for our new Children’s Hospital in Leeds which will be built in 2025.”