The Prime Minister says the restrictions preventing the return of recreational can not yet be lifted.
He subsequently confirmed pubs, restaurants, hotels, cinemas and hairdressers can reopen from 4 July providing they are "COVID-secure".
Clark said half of summer cricket has been lost and asked the PM if the game could be resumed now.
Boris Johnson used a House of Commons speech to exaggerate the number of schools with returning pupils, the government's own data suggests.
"We have been round it many times with our scientific friends. At the moment, we are still working on ways to make cricket more covid-secure, but we can not change the guidance yet".
It prompted a statement from the European Central Bank, who believe the game is as safe as any others, and are looking for play to resume in July.
The ECB subsequently issued a statement stressing they wanted to work with Johnson's government to get amateur cricket going again "on or around 4 July" and that as, a non-contact sport, it can be played "as safely as many other activities being now permitted".
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He added that customers will be limited to table service and only allowed to visit with one other household. We know this pandemic has been particularly hard for people working in hairdressing and hospitality.
On Tuesday evening, the European Central Bank - who have been at pains to stress they have engaged in regular dialogue with the government throughout the coronavirus crisis - expressed a preferred date for amateur cricket's return for the first time: "on or around July 4".
The government explained that this is because it wants to make sure the risk of transmission will not be increased as "we begin to reopen the economy".
"We can confirm that any guidance we share with the game will include directions on how to mitigate any risk from handling the ball as we continue to prioritise the health and safety of the cricket family in all our decision-making".
"We can not lift all the restrictions at once, so we have to make hard judgments, and every step is scrupulously weighed against the evidence", Johnson said.
Labour leader Sir Keir Starmer said he welcomed the statement overall, adding " I believe the government is trying to do the right thing and in that I support them".
The ECB have already announced that county cricket is postponed until August 1, at the earliest.
England are due to take part in the first major worldwide match since lockdown when they face the West Indies in the opening Test of a three-match series behind closed doors at Southampton on 8 July.