Cllr Derek Bastiman said that the furore over Mr Cummings was distracting from the messages around the coronavirus pandemic.
Cllr Bastiman added that he had been contacted by residents who wanted to "move on" from the story.
Yesterday, Mr Cummings gave a press conference defending his decision to drive more than 250 miles from London to Durham in March to self-isolate with his family.
He also said that while there he took a trip with his family to nearby Barnard Castle to test his eyesight to see if he could make the trip back down to London.
Mr Cummings said he believed he had acted "reasonably" and within the law and refused to apologise for his actions.
Cllr Bastiman said that Mr Cumming's actions had been guided by a wish to protect his family.
He said: "He did what he thought was the best thing for his wife and his child and under the circumstances, I think people understand that.
"The majority of the people who have been in contact with me are concerned that it is diverting attention away from more important issues and taking too much air time.
"I think it has been blown out of all proportion, he did what was right and proper in the circumstances for his family."
Cllr Bastiman added that he did not think that it would detract from the messaging around the virus.
He added: "I think people can see through [the reporting].
"The emails I have had from people say that it's time to move on, let's be concerned about getting our high street open, getting people back to work where they want to be and let's get people spending to boost the economy."
In contrast, Scarborough and Whitby's Conservative MP Robert Goodwill has called Mr Cummings position "untenable" and that he should be sacked.
Mr Goodwill said he had had more than 500 emails, an "unprecedented" number, over the weekend from constituents.
In response to emails before Mr Cummings statement, Mr Goodwill said he thought Boris Johnson's advisor should be "removed from his post" and today (Tuesday) said he stood by those claims.
He said: "I've not had any emails or contact since the statement from people saying that what he said had changed their minds."
Mr Goodwill said many people had made sacrifices to stop the spread of the virus and that Mr Cummings had "fallen short of the standards we expect of every individual, never mind a person who should be setting an example".
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