We consider a skew-symmetric n-ary bracket on the polynomial algebra K[x₁, . . .,x_n,x_n+1] (n ≥ 2) over a field K of characteristic zero defined by {a₁, . . .,a_n}=J(a₁, . . .,a_n,C), where C is a fixed element of K[x₁, . . .,x_n,x_n+1] and J is the Jacobian. If n = 2 then this bracket is a Poisson bracket and if n ≥ 3 then it is an n-Lie-Poisson bracket on K[x₁, . . .,x_n,x_n+1]. We describe the centre of the corresponding n-Lie-Poisson algebra and show that the quotient algebra K[x₁, . . .,x_n,x_n+1]/(C-λ), where (C-λ) is the ideal generated by (C-λ), 0 ≠ λ ∈ K, is a simple central n-Lie-Poisson algebra if C is a homogeneous polynomial that is not a proper power of any non-zero polynomial. This construction includes the quotients P(𝔰𝔩₂(K))/(C-λ) of the Poisson enveloping algebra P(𝔰𝔩₂(K)) of the simple Lie algebra 𝔰𝔩₂(K), where C is the standard Casimir element of 𝔰𝔩₂(K) in P(𝔰𝔩₂(K)). It is also proven that the quotients P(𝕄)/(C-λ) of the Poisson enveloping algebra P(𝕄) of the exceptional simple 7-dimensional Malcev algebra 𝕄 are central simple.