Brooklyn 5-year-old Kyan Dickerson only grew more ill as the city Housing Authority fought to keep from cleaning the lead out of his family’s apartment, according to new documents obtained by The Post.

“He probably wouldn’t have been sick” if NYCHA had addressed the issue the first time lead was found in the Red Hook East public-housing apartment, the boy’s mother, Sherron Paige, told The Post on Sunday.

She said her son’s lead levels had finally dropped to 6 micrograms when he went to the doctor six months ago. But the effects of the lead exposure remain clear, she said.

“He still has his temper problem,” Paige said. “When he gets mad, he gets frustrated, he starts breaking stuff.”

Paige is among a group of parents suing the New York City Housing Authority in Manhattan federal court over lead contamination in their homes.

Kyan’s doctor ran blood tests that found elevated levels of lead in the blood of the then-4-year-old in July 2017. The Brooklyn boy registered 12 micrograms per deciliter, above the city’s then-poisoning threshold of 10 micrograms.

That triggered an August visit from a city Health Department inspector armed with an X-ray gun who found four lead hot spots in the apartment.

The inspector “observed a huge hole on wall #4 of the foyer and wall #2 of the kitchen with severe chipping paint resulting in gross dust from the paint,” documents show.

NYCHA sent its own employee to collect chip samples and had the paint analyzed by an independent lab, which found that there was some lead but not at supposedly highly toxic levels.

Critics gripe that allowing NYCHA to gather its own samples was hardly transparent.

“It’s totally self-serving,” said Matt Chachere, a lawyer and lead expert with the nonprofit Northern Manhattan Improvement Corporation. “You don’t need to have a degree or anything to see the flaw in that methodology.”

But based on the lab’s results, city health officials nixed its order for a clean-up of the family’s apartment. Still, the department told NYCHA that it “strongly recommends that you bring all areas with peeling paint to an intact condition.”

Health inspectors returned to the apartment 17 days later to conduct another check — this time using wipes to detect lead dust, the report shows. By that point, Kyan’s lead poisoning had worsened to at least 15 micrograms of lead, according to the paperwork.

The majority of the wipes came back positive and NYCHA was ordered in September to clean the apartment, which the authority completed within a week.