"On behalf of NHS [National Health Service] breast screening services, we apologise to the women affected and we are writing to them to offer a catch-up screening appointment", PHE Deputy Medical Director Jenny Harries said in a statement.
Breast cancer screening in England is offered once every three years to women aged 50 to 70.
The mistakes first came to the attention of health chiefs in January after nearly a decade of errors and Mr Hunt said "serious failure" of a "computer algorithm" was to blame.
While it's well-known that a lump in the breast could be a sign of the disease, there are many lesser-known symptoms that women (and indeed, men) should be aware of, as treatment is more successful if it's detected early.
At the time of the discovery, Hunt said, Public Health England advised him that the public should not be told to allow time for remedial measures to be put in place.
He said any woman who wanted a mammogram would get one within six months, and it was a priority to make sure additional scans did not cause any delays in the screening programme for other women.
- What happens if I am affected?
"He said there could be 135 to 270 women who ". had their lives shortened as a result" of the failure.
Professor Helen Stokes-Lampard said: 'We are shocked to learn that hundreds of thousands of women in England have missed out on their opportunity for breast screening - and the implications for Global Positioning System and our teams will potentially be significant, as patients seek reassurance and to find out where they go from here.
'Free press makes for a stronger democracy'
Let us celebrate the freedom of press with dignity and with gratitude towards these journalists on this World Press Freedom Day . Prime Minister Narendra Modi on Thursday applauded the media on the World Press Freedom Day and vouched for a free Press.
The records of the women participating had a "flag" on the NHS system when the trial began in 2009, resulting in them not receiving any more routine invitations for screening.
AgeX was created to examine whether screening should be extended from the current age range of 50-70 to include those aged between 47 and 49, and between 71-73.
Poor awareness of screening and breast cancer symptoms are among the reasons why the figures are so high among black women.
"PHE will be directly contacting all women affected - we'd urge against panic and encourage anyone concerned that they have missed their screening invitations to contact the PHE helpline directly for further advice".
Hunt agreed with Miller and told the Commons that he believed "a single IT mistake" led to the issue.
Baroness Delyth Morgan, chief executive at Breast Cancer Now, welcomed the independent inquiry.
He said the error was a serious failure of the screening program, which is run by Public Health England and tests are carried out by NHS hospitals.
"We must also recognise that there may be some who receive a letter having had a recent terminal diagnosis".
Shadow Health Secretary Jon Ashworth welcomed the probe and called for reassurances from the Government that no more patients would be affected by IT failures.