Generalized harmonic number sums and quasisymmetric functions

We express some general type of infinite series such as ∑^∞ₙ₌₁F(Hₙ⁽ᵐ⁾(z),Hₙ⁽²ᵐ⁾(z),…,Hₙ( m)(z)) (n+z)s₁(n+1+z)s₂⋯ (n+k-1+z)sₖ, where F(x₁,…,x₎∈ Q[x₁,…,x_], Hₙ⁽ᵐ⁾(z)=∑ⁿⱼ₌₁1/(j+z)ᵐ, z∈ (-1,0], and s₁,…,sₖ are nonnegative integers with s₁+⋯+sₖ≥ 2, as a linear combination of multiple Hurwitz zeta functions and some speical values of Hₙ⁽ᵐ⁾(z).