Future wireless access networks will support simultaneously a large number of devices with heterogeneous service requirements. These include data rates, error rates, and latencies. While there exist achievable rate and capacity results for Gaussian broadcast channels in the asymptotic regime, the characterization of second-order achievable rate regions for different blocklength constraints are not available. Therefore, we investigate a two-user Gaussian broadcast channel (GBC) with heterogeneous blocklength constraints under a maximal input power constraint and an average error probability constraint. Unlike the traditional GBC where two users have the same blocklength constraints, here the user with higher output SNR has a shorter blocklength constraint. We show that with sufficiently large output SNR, the stronger user can invoke the technique named early decoding (ED) to decode the interference. Then the successive interference cancellation (SIC) can proceed. This leads to an improved achievable rate region compared to the state of the art. To achieve it, we derive an explicit lower bound on the necessary number of received symbols for a successful ED, using an independent and identically distributed Gaussian input. A second-order rate of the weaker user who suffers from an SNR change due to the heterogeneous blocklength constraint, is also derived. We then formulate the rate region of the considered setting with individual and also sum power constraints and compare to that of the hybrid non-orthogonal multiple access (HNOMA) scheme. Numerical results show that ED has a larger rate region than HNOMA partly when the gain of the better channel is sufficiently larger than the weaker one. Under the considered setting, about 7-dB SNR gain can be achieved. This makes ED with SIC a promising technique for future wireless network.