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Downing Street says this is “false” and cabinet minister Grant Shapps states Mr Cummings – who has actually been seen going into No 10 – will not give up.
Labour has called for an urgent questions into the accusations, while federal government ministers rallied around Mr Cummings on Saturday and defended his conduct.
But, she stated, the “integrity” of public health guidance “should come initially” and she advised Mr Cummings to resign.
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The 2 papers have now reported witnesses saw Mr Cummings in Barnard Castle, more than 25 miles from Durham, on 12 April.
On 14 April, he was seen in London. According to a witness, he was found again near Durham in Houghall Woods on 19 April.
Mr Cummings is yet to publicly react to the new claims, however the Sunday Telegraph reports he informed Downing Street he left Durham on 13 April, which the claim he made a second trip from London was “totally false”.
Transport Secretary Grant Shapps told the BBC’s Andrew Marr that the claims of a 2nd journey was “incorrect”.
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However there are growing calls from backbench Tory MPs for Mr Cummings to consider his position.
Tory Sir Roger Windstorm said there “can not be one law for the prime minister’s staff and another for everybody else”.