The Poisson structure arising in the Hamiltonian approach to the rational Gaudin model looks very similar to the so-called modified Reflection Equation Algebra.  Motivated by this analogy, we realize a braiding of the mentioned Poisson structure, i.e. we introduce a ”braided Poisson” algebra associated with an involutive solution to the quantum Yang-Baxter equation. Also, we exhibit another generalization of the Gaudin type Poisson structure by replacing the first derivative in the current parameter, entering the so-called local form of this structure, by a higher order derivative.  Finally, we introduce a structure, which combines both generalizations.  Some commutative families in the corresponding braided Poisson algebra are found.