A deputy director with Chirlane McCray’s embattled $1 billion mental-health project ThriveNYC admitted in an e-mail that her group “mishandled” a seminar allegedly scuttled over its ties to Blue Lives Matter.

The training session was for first-responders and supposed to feature counselors from the costly initiative founded by Mayor de Blasio’s wife.

But Thrive pulled out at the last minute over the involvement of the pro-cop group Blue Lives, city Councilman Joe Borelli exclusively told The Post this week — a claim hotly denied by the mayor.

Yet a damning e-mail shows Thrive Deputy Director Sarah Solon wrote Borelli’s chief of staff, Frank Mascia, on Aug. 9 — conceding, “Frank, I think our office mishandled this.”

WNYC radio host Brian Lehrer pushed the mayor Friday about the e-mail, sending Hizzoner into a tizzy.

“I don’t know the details of what some mid-level person said or didn’t say,” de Blasio seethed to Lehrer as he continued to call the criticism of his wife’s pet program “lies.”

The Democratic mayor acknowledged that “there was confusion about whether it was a public event or a by-invitation-only” session but insisted that either way, the issue “could have been fixed” in time.

Borelli, a Republican whose Staten Island office was sponsoring the event, shot back to The Post on Friday, “If the mayor doesn’t know what his staff is doing or saying, he may want to rethink calling me a liar.”

The councilman said Thrive clearly yanked its offer to provide therapists for the scheduled Aug. 19 mental-health seminar because Blue Lives Matter was recruiting police to sign up to attend.

Blue Lives Matter, formed in the aftermath of the fatal 2014 ambush of NYPD officers Rafael Ramos and Wenjian Liu, is regarded as a controversial group by some for its juxtaposition to Black Lives Matter.

“When you say you don’t want to be associated with the group that is hosting the event and bringing the police officers, then OK, we’re not going to have the event,” Borelli explained of the decision not to move forward with it.

At the time, seven NYPD police officers had committed suicide this year. Before another week was out, that tragic number jumped to nine.

E-mails between City Hall staff and Borelli’s office show the administration was actually at first on board with Blue Lives Matter participating.

On June 27, Mascia wrote to Thrive staffer Charlie Aidinoff, “How about 7/25 we do a training for constituents with a focus on first responders & partner with blue lives matter,” according to a copy of the e-mail obtained by the Post.

Shauna Stribula, who is with the Mayor’s Office of Community Affairs, was on the e-mail chain and replied the same day, “Works for us!”

The training was eventually scheduled for Aug. 19.

Then Aug. 9, Aidinoff’s supervisor, Solon, saw a copy of a flier advertising the training with the Blue Lives Matter logo near the top and called Mascia, the aide said.

She “was obviously uncomfortable with the flier. She didn’t want their logo used,” Mascia said, recalling their phone conversation.

Shortly after, Solon sent the e-mail that said, “I think our office mishandled this. Let me know if you’d like to speak.”

They never had a further conversation about the matter. Borellis said it was because he felt he’d had enough of being jerked around by Thrive and shunning Blue Lives Matter was a deal-breaker.

Solon had a different take on her conversation with Mascia.

“I called to tell him about the public vs. private training issue, and he got frustrated, cursed and hung up the phone,’’ Solon told The Post. “I e-mailed because I thought the conversation could have gone better and wanted to reconnect. I didn’t hear from him again.”