The prime minister was passed the device so he could view the image of Jack Williment-Barr, who had to lie on a coat for four hours with suspected pneumonia because of a lack of beds.
Sarah Williment, the boy's mother, told the Daily Mirror that she would now be voting Labour as a result of the harrowing experience.
"We are very sorry that the treatment room had only chairs. This falls below our usual high standards, and for this we would like to sincerely apologise to Jack and his family".
Last week Jack with vomiting, diarrhoea, cough and fever was brought to the hospital by ambulance.
"He kept asking to lay down".
Statistics from the first week of December shows 5,171 of the 103,099 patients seen in A&E were stuck there for more than 12 hours and only 68% were seen within the four-hour target time.
"There was no bed for him in A&E and there was no bed for him on the ward, so he just had to sleep on the floor".
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The boy in the picture was eventually diagnosed with flu and tonsillitis and was allowed to be taken home after waiting for several hours in various parts of the hospital in Leeds, northern England.
Mrs Williment praised medical staff but said: "I am frustrated about the system and the lack of beds, which I am presuming is due to a lack of funding to the NHS to deliver the services that are required".
He said: "Boris Johnson should personally apologise to Jack and his family".
"Refusing to look at a photo will not help our NHS", Corbyn tweeted Monday.
"Don't give this disgrace of a man five more years of driving our NHS into the ground".
Asked about accusations he was using the NHS as a political football on BBC Breakfast he said: 'It's an example of what's happening in our NHS.
"What we are doing is supporting the NHS and on the whole, I think patients in the NHS have a much, much better experience than this poor kid has had".
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Liberal Democrat Luciana Berger said: "This young boy's experience is absolutely shameful and deeply upsetting".
During an interview on Monday, Joe Pike, a political correspondent with ITV News, attempted to show Johnson a picture of the boy on his cell phone, but the prime minister repeatedly refused to look and at one point grabbed Pike's phone and put it in his pocket.
"I'm afraid the short answer to that question is that I haven't got the foggiest idea", Mr Johnson responded.
Noting that the clip of the interview has more than 10 million views, Guardian columnist and Labour supporter Owen Jones said "that means millions of people still have not seen the horrific lack of empathy Boris Johnson has towards a 4 year old victim of his policies".
Prime Minister Boris Johnson snatched a reporter's phone after the reporter tried to show him a picture of a 4-year-old boy who was forced to sleep on a hospital floor.
When challenged again, Johnson took the phone out, looked at it and said: "It is a awful, bad photo, and I apologise obviously to the family and all those who have awful experiences in the NHS but what we are doing is supporting the NHS".
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