"We have now got cases where patients who should be being treated, we can't treat them because they can't get access to a test", he told Sky News television in an interview.
Mr Hancock said prioritisation was "a choice that we must make". "It seems to me there'll be extreme cases where people can't get to test locations within that radius, but that doesn't mean that Public Health England are not working night and day to boost capacity", she said. "I think there is a surge in demand (and) I think our stated capacity is very different from actually how many tests can be run in a given day", he said.
NHS staff absences have started to increase notably in parts of the country as staff and their family members struggle to access coronavirus tests, forcing them to remain in self-isolation in the absence of a test for themselves or their close family.
The British Medical Association has said the government should focus on the current testing system - rather than its enhanced "Operation Moonshot" proposal.
"I do not shirk from decisions about prioritisation", Mr Hancock said.
Scottish first minister Nicola Sturgeon said yesterday she was seeking "urgent discussions" with the United Kingdom government over delays in people receiving their test results. They're not always comfortable, but they are important.
"The problem is that NHS trusts are working in the dark - they don't know why these shortages are occurring, how long they are likely to last, how geographically widespread they are likely to be and what priority will be given to healthcare workers and their families in accessing scarce tests".
Acute clinical care is the top priority, with social care next on the list and now receiving more than 100,000 tests a day.
Britain's testing system for COVID-19 was creaking on Tuesday as a bottleneck prevented people including medics from getting a test, with the government saying it may take weeks to resolve the problem.
"We have seen a sharp rise in people coming forward for a test, including those who are not eligible", Mr Hancock said.
The Department of Health said there were tests available and urged people to keep trying.
A government website providing statistics on coronavirus testing shows that the number of tests processed each day, which was around 200,000 last week, is by far exceeded by the existing estimated total testing capacity reported by the laboratories, which amounted to over 374,000 tests per day as of September 10.
The testing situation was discussed at the Cabinet meeting on Tuesday.
Britain advises those showing symptoms should get a test though it says the system has been burdened by people with no symptoms asking for tests, while some schools have demanded any ill students get a test or stay away for 14 days.
She added: "Our members, and especially in general practice, are concerned that their staff are having to self-isolate unnecessarily due to delays and difficulties in access to tests for themselves and their families - and that some are having to wait longer than expected for the results - which is putting a strain on the services they can provide".
"A month ago, they had spare capacity in testing, significant spare capacity, but I think what has been underestimated was the speed at which the second wave would arrive, but also the pressure put on the system from children returning to school, and the testing demands associated with that, and people increasingly out and about".
Professor Alan McNally, director of the institute of microbiology and infection at the University of Birmingham, told the BBC there were clearly underlying issues which nobody wants to tell us about, together with increased need for testing.
While the number of people in hospital with Covid-19 was relatively low, the source said "the direction of travel is only going to go one way, in my view between now and the winter, which is up".
They said that whereas younger people were most affected in recent weeks, this has now translated into older age groups catching the virus.