Dynamic programming with housing consumption and labor
$$V(b_{t-1})=\max\limits_{C_{t},H_t,N_t} \Bigg\{\ln C_t+j\ln H_t-\dfrac{(N_t)^\eta}{\eta} +\beta E_tV(b_t) \Bigg\}$$

so 

$$\beta E_tV(b_{t})=\beta E_t\left[\max\limits_{C_{t+1},H_{t+1},N_{t+1}} \Bigg\{\ln C_{t+1}+j\ln H_{t+1}-\dfrac{(N_{t+1})^\eta}{\eta} +\beta E_{t+1}V(b_{t+1}) \Bigg\}\right]$$

But forwarding the budget constraint 

$$C_t + b_t +q_t(H_t-H_{t-1})=\dfrac{R_{t-1} b_{t-1}}{\pi_t}+w_t N_t+F_t$$

we get after re-arranging, 

$$C_{t+1} =- b_{t+1} -q_{t+1}(H_{t+1}-H_{t})+\dfrac{R_{t} b_{t}}{\pi_{t+1}}+w_{t+1} N_{t+1}+F_{t+1}$$

So $\beta E_tV(b_{t})$ is also a function of $H_t$ through $C_{t+1}$.