This paper investigates streaming codes for three-node relay networks under burst packet erasures with a delay constraint $T$. In any sliding window of $T+1$ consecutive packets, the source-to-relay and relay-to-destination channels may introduce burst erasures of lengths at most $b_1$ and $b_2$, respectively. Let $u = \max\{b_1, b_2\}$ and $v = \min\{b_1, b_2\}$. Singhvi et al. proposed a construction achieving the optimal rate when $u\mid (T-u-v)$. In this paper, we present an extended delay profile method that attains the optimal rate under a relaxed constraint $\frac{T - u - v}{2u - v} \leq \left\lfloor \frac{T - u - v}{u} \right\rfloor$ and it strictly cover restriction $u\mid (T-u-v)$. %Furthermore, we demonstrate that the optimal rate for streaming codes is not achievable when $0< T-u-v<v$ under the convolutional code framework.