Expectation of product of hermitian matrix
You could define a diagonal matrix $\mathbf{D}$ with diagonal elements $d_{ii}=a_i+b_i$. Then vector $\mathbf{h}$ can be written as

$$\mathbf{h}=\mathbf{D}\mathbf{c}$$

and the expectation is

$$\mathbb{E}[\mathbf{h}\mathbf{h}^H]=\mathbf{D}\mathbb{E}[\mathbf{c}\mathbf{c}^H]\mathbf{D}^H$$

(I assume that $\mathbf{a}$ and $\mathbf{b}$ are deterministic and $\mathbf{c}$ is random, because you said you wanted the expectation in terms of $\mathbf{c}$.)