The City Council moved Wednesday to join a lawsuit that seeks to force Mayor Eric Adams to implement a new set of laws designed to make it easier for low-income New Yorkers to access the city’s rental assistance program. The Manhattan Supreme Court lawsuit was first filed last week by four New Yorkers at risk of homelessness who’d become eligible for vouchers via the program, CityFHEPS, if the mayor allowed the laws to take effect. The laws would expand access to the program, which subsidizes rent on open market apartments, by increasing the income eligibility cap and eliminating a rule that r…