In this paper an $H$-generalized Cauchy equation \[ S(t+s)C=H(S(s),S(t)) \] is considered, where $\{S(t)\}_{t\geq0}$ is a one parameter family of bounded linear operators and $H\colon B(X)\times B(X)\to B(X)$ is a function. In the special case, when $H(S(s),S(t))=S(s)S(t)+D(S(s)-T(s))(S(t)-T(t))$ with $D\in B(X)$, solutions of H-generalized Cauchy equation are studied, where $\{S(t)\}_{t\geq0}$ is a $C$-semigroup of operators. Also a similar equations are studied on $C$-cosine families and integrated $C$-semigroups.