It follows critical findings recorded at the end of a three-week inquest into the death of Chris Beardshaw on December 30, 2013.
Jurors at Wakefield Coroners Court heard how Mr Beardshaw made around 40 cuts to his arm using the plastic knife provided for his meal, triggering a number of actions to address concerns for his safety.
However, the measures taken by staff at the prison in Armely failed to stop Mr Beardshaw from taking his life later that day.
A Ministry of Justice spokesman said: "“This is a tragic case and our thoughts are with Chris Beardshaw’s family and friends.
“The safety and welfare of people within our custody is our top priority but we recognise that there were significant failings in his care.
“We will carefully consider the findings of the inquest, and make sure all necessary measures are in place to better support offenders.”
The inquest heard there had been no previous concerns about Mr Beardshaw's mental health, but he had become distressed about the condition of the cell he was placed in on the day he died.
After Mr Beardshaw cut himself, staff opened an Assessment, Care in Custody and Teamwork (ACCT) Review to note concerns about his mental health.
They also removed furniture and replaced his clothes with what officers believed was anti-tear clothing.
But the jury noted a number of issues on Thursday when they returned a narrative verdict which set out the sequence of events leading up to Mr Beardshaw's death.
They highlighted concerns around the level of staff training, the quality of ACCT observations and communication between officers, and evidence that one officer had failed to check on Mr Beardshaw as recorded.
The findings prompted Coroner Jonathan Leach to say that he would be preparing a Prevention of Future Deaths report.
INQUEST, a charity which focuses on the issue of deaths in custody, said the jury's verdict added to existing concerns about the high number of self-inflicted deaths in prisons in England and Wales.
And in a statement issued yesterday, Mr Beardshaw's family said they hoped action would be taken to address the issues highlighted.
It read: "Chris was so optimistic and working hard to get his life on track. We do not believe he wanted to die; he just wanted someone to listen to him, it was a cry for help.
"After listening to three weeks of evidence we believe that there were failings in Chris’s care and we hope that the prison acts upon those issues brought to light in the inquest to prevent other families experiencing the heartache we have.”
There have been a further 10 self-inflicted deaths at the prison since Mr Beardshaw died on December 30, 2013.
It is second highest rate in any prison in England and Wales, with only HMP Woodhill recording a higher number.