Letby was found guilty of murdering seven babies and attempting to kill six others following a 10-month trial.
The former nurse deliberately injected babies with air, force-fed others milk and poisoned two with insulin while working at the Countess of Chester Hospital.
Letby, the most prolific child serial killer in modern British history, was given a whole-life sentence in August.
Nottingham University Hospitals’ Trust board has discussed how it can ensure that a similar case cannot happen within the trust.
He told the trust board on September 14: “The shockwaves sent through the system by what happened in the Lucy Letby case is significant and we must use it as motivation to ensure something similar couldn’t happen here.”
Since the case, medics who worked at the hospital in Chester at the time have said they felt ignored or discouraged when they tried to warn senior managers of earlier suspicions about Letby.
To avoid a similar risk in future, the Nottingham trust says it wants to create a “speak up culture” where staff and patients can raise concerns.
The trust says there are already three full-time ‘Freedom to Speak Up Guardians’ in place, who raise awareness about how to talk.
It is also rolling out a ‘rapid respond’ approach to incidents, so staff can take action “from the earliest opportunity”.
The organisation, which runs the Queen’s Medical Centre and City Hospital, has recently received a ‘requires improvement’ rating from the healthcare watchdog for its maternity and leadership.
This was an improvement on the previous ‘inadequate’ ratings.
The watchdog says the chair and chief executive had been “instrumental in empowering people to speak up” – but “staff did not always feel able to raise concerns without fear of retribution”.
Mr May said he had received a letter from NHS England following the outcome of the trial of Letby, which the trust must comply with.
The letter “outlines the decisive steps already taken towards strengthening patient safety monitoring and details specific measures NHS leaders and Boards must have in place as a matter of urgency”.
Mr May said: “In my view, the ability of colleagues to speak up and patients to speak up when something is wrong is entirely a feature of leadership and culture.
“We are working really hard to improve culture and there are signs that we are doing that.
“It’s a big organisation and we know we’ve got an awful lot to do.
“If we can improve the culture and ensure that everybody here feels equally valued, it is less likely that the appalling crimes by Lucy Letby will happen at NUH.
“There are some notable things we are doing. We have recruited a director of governance [Gilbert George].
“Since Gilbert has joined us there is much better closer working between senior colleagues and people who report to them.
“We have recently had the benefit of a CQC inspection of our governance and by and large, they saw some positive signs that we have opened up the culture and are listening to people more.
“We all know we’ve got a lot further to go before we can be satisfied.”
Chief Nurse Michelle Rhodes added: “Although Letby’s crimes were in a neonatal unit, this could happen in any service.
“We need to make sure we apply learning across the organisation.”