You can write $h(s)=\int_0^Tf(t)dtH(s)$ then your expectation could be written as $E[ \int_{o}^{T} h(s) dW(s)]$. This integral is $0$ if you can prove $(\int_{o}^{t} h(s) dW(s)) $ to be a martingale, for example $E(\int_{o}^{T} h^2(s) ds)<+\infty$.