Mayor de Blasio on Monday pledged more homeless outreach after a vagrant killed four other street people, but critics say the system needs an overhaul — starting with the first lady’s “failure” of a mental health initiative.

“The city is spending hundreds of millions of dollars on the very necessary work of providing comprehensive mental-health care, but Thrive must do better to ensure these programs are reaching those most in need of help,” city Comptroller Scott Stringer told The Post, referring to ThriveNYC, the $850 million program started by Hizzoner’s wife, Chirlane McCray, to combat mental-health issues.

DJ Jaffe, director of the Mental Illness Policy Org, called the man accused in Saturday’s bloodbath in Chinatown “likely an example of what happens when you have a mayor and first lady who flood mental-health programs that refuse to serve the most seriously mentally ill with funds.”

Just 12 percent of Thrive’s $250 million annual budget helps people with serious mental illness, he said.

Councilman Joe Borelli (R-SI) added that the rampage “highlights not one but two de Blasio administration failures: the failure to address homelessness in general and the failure to address mental illness through ThriveNYC.”

One of the main aspects of McCray’s initiative is to include training for first responders so they can better identify patients with mental-health issues — something even City Hall analysts say the program has fallen short on.

De Blasio said on Monday that the city was deploying Thrive and other Health Department outreach teams to Chinatown — and creating emergency pop-up crisis centers to better help the homeless.

But no homeless people showed up on Monday at one center, at 62 Mott St., during its hours from 1:30 p.m. to 4:30 p.m.

Omar Muhammad Robinson, who is homeless and got a Thrive brochure outside the center, told The Post: “It’s just talking. We’re still homeless.

“I haven’t washed my clothes in months. I haven’t had a shower. Nothing has been solved. Nothing is getting done. It just seems like a knee-jerk reaction to a horrific murder.”