Let X and Y be vector spaces. It is shown that a mapping $f:X{ ightarrow}Y$ satisfies the functional equation ${ddag}$ $$2df(frac{x_1+{sum}_{j=2}^{2d}(-1)^jx_j}{2d})-2df(frac{x_1+{sum}_{j=2}^{2d}(-1)^{j+1}x_j}{2d})=2sum_{j=2}^{2d}(-1)^jf(x_j)$$ if and only if the mapping $f:X{ ightarrow}Y$ is additive, and prove the Cauchy-Rassias stability of the functional equation (${ddag}$) in Banach modules over a unital $C^*$-algebra, and in Poisson Banach modules over a unital Poisson $C^*$-algebra. Let $mathcal{A}$ and $mathcal{B}$ be unital $C^*$-algebras, Poisson $C^*$-algebras, Poisson $JC^*$-algebras or Lie $JC^*$-algebras. As an application, we show that every almost homomorphism $h:mathcal{A}{ ightarrow}mathcal{B}$ of $mathcal{A}$ into $mathcal{B}$ is a homomorphism when $h(d^nuy)=h(d^nu)h(y)$ or $h(d^nu{circ}y)=h(d^nu){circ}h(y)$ for all unitaries $u{in}mathcal{A}$, all $y{in}mathcal{A}$, and n = 0, 1, 2, ${cdots}$. Moreover, we prove the Cauchy-Rassias stability of homomorphisms in $C^*$-algebras, Poisson $C^*$-algebras, Poisson $JC^*$-algebras or Lie $JC^*$-algebras, and of Lie $JC^*$-algebra derivations in Lie $JC^*$-algebras.