Jackie Doyle-Price described the situation as an “embarrassing episode” for the government.
The Prime Minister’s chief adviser drove 260 miles from London to Durham during the coronavirus lockdown on Friday, 27 March.
At the time, Mr Cummings suspected his wife had Covid-19 and the following morning he came down with coronavirus symptoms.
Government advice at the time told people to stay at home if one or more person was infected inside their household.
Ms Doyle-Price said that following that guidance “has come at great personal cost” to many people.
She said: “Some people have lost their livelihoods. Some people have lost loved ones and been unable to say goodbye.
“In making real sacrifices in order to obey the Government’ instructions it is quite understandable that there is anger that Government figures have not done the same.
“When a Government is tainted with a ‘do as I say, not as I do’ attitude to the public, it loses its moral authority.”
During a press conference on Monday, Mr Cummings said he does not regret making the journey and believes he acted “reasonably”.
The senior adviser also claimed he drove from Durham to Barnard Castle to “test his eyesight” on Sunday, 12 April.
Mr Cummings said he went on the test trip because he was worried he would not be able to safely drive all the way home to London as his eyesight was affected by coronavirus.
“I have witnessed with dismay as the Government machine has sought to defend Mr Cummings,” said Ms Doyle-Price.
“In doing so it has endorsed behaviour which is clearly not consistent with what the Government expects each of us as citizens to do.
“At a time when the public is looking to Government for leadership through this dangerous pandemic, this handling has been ill-judged.”
“Frankly this has been a very embarrassing episode for the Government. To bring this sorry chapter to an end I am afraid Mr Cummings has to go.”
More than 30 Conservative MPs have sent similar messages to the Government, calling for Mr Cummings to go.
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