In this paper, we consider several convolution sums, namely, $mathcal{A}_i(m,n;N)$ ($i=1,2,3,4$), $mathcal{B}_j(m,n;N)$ ($j=1,2,3$), and $mathcal{C}_k(m,n;N)$ ($k=1,2,3,{cdots},12$), and establish certain identities involving their finite products. Then we extend these types of product convolution identities to products involving Faulhaber sums. As an application, an identity involving the Weierstrass ${wp}$-function, its derivative and certain linear combination of Eisenstein series is established.