The Independent Police Complaints Commission (IPCC) called for an "urgent review" into how officers deal with vulnerable people.
The release of the IPCC report comes after an inquest into the death of Robert Cox - who was stabbed by Derek Hancock in 2013 - earlier this month.
The force said it now worked "side by side with mental health professionals".
Hancock pleaded guilty to the manslaughter of 24-year-old Robert Cox on the grounds of diminished responsibility.
The 42-year-old was sentenced to life imprisonment with a minimum term of four years and nine months in December 2014.
Described as "deluded and psychotic" in court, Hancock had repeatedly made false allegations that Mr Cox and other residents at supported accommodation at Egerton Road had made sexual advances towards him.
The inquest found Mr Cox was unlawfully killed and led to the publication of the IPCC's recommendations, which include:
Two police officers and two call handlers were cleared of any wrongdoing by an IPCC investigation.
A third call handler who told Hancock, who was threatening to take the law into his own hands minutes before killing Mr Cox, that it was "entirely up to you" was sacked by the force.
IPCC associate commissioner Guido Liguori said: "These are tragic circumstances and I offer my sincere condolences to the family and friends of Robert Cox.
"The training and support provided to staff to engage and support people with mental health issues was inadequate, and it is concerning that the calls made by Mr Hancock from Egerton Road in the months leading up to the incident were neither linked nor marked as being made from a home with vulnerable residents."
An Avon and Somerset police spokesman said: "We've already established a mental health triage system within our call centre, which sees call handlers working side by side with mental health care professionals.
"Since this was introduced in September, hundreds of people have benefitted from this service."
Training for Avon and Somerset police officers in dealing with mentally ill people is "inadequate", a report says.