Can I solve the partial derivative (dG/dH)s ? Does it make sense?
$$dH=TdS+VdP$$
At constant S, $$dH=VdP$$
$$dG=-SdT+VdP$$
If S is constant, then $$\left(\frac{\partial S}{\partial T}\right)_PdT+\left(\frac{\partial S}{\partial P}\right)_TdP=0$$But,$$\left(\frac{\partial S}{\partial T}\right)_P=\frac{C_p}{T}$$and$$\left(\frac{\partial S}{\partial P}\right)_T=-\left(\frac{\partial V}{\partial T}\right)_P$$So, $$dT=\frac{T}{C_p}\left(\frac{\partial V}{\partial T}\right)_PdP$$and$$dG=\left[V-S\frac{T}{C_p}\left(\frac{\partial V}{\partial T}\right)_P\right]dP$$Dividing by dH then gives:
$$\left(\frac{\partial G}{\partial H}\right)_S=1-\frac{\alpha T}{C_p}S=1+\frac{(G-H)\alpha}{C_p}=1+\frac{\alpha T}{C_p}\left(\frac{\partial G}{\partial T}\right)_P$$