Class-incremental learning (CIL) aims to train classifiers that learn new classes without forgetting old ones. Most CIL methods focus on balanced data distribution for each task, overlooking real-world long-tailed distributions. Therefore, Long-Tailed Class-Incremental Learning (LT-CIL) has been introduced, which trains on data where head classes have more samples than tail classes. Existing methods mainly focus on preserving representative samples from previous classes to combat catastrophic forgetting. Recently, dynamic network algorithms frozen old network structures and expanded new ones, achieving significant performance. However, with the introduction of the long-tail problem, merely extending task-specific parameters can lead to miscalibrated predictions, while expanding the entire model results in an explosion of memory size. To address these issues, we introduce a novel Task-aware Expandable (TaE) framework, dynamically allocating and updating task-specific trainable parameters to learn diverse representations from each incremental task, while resisting forgetting through the majority of frozen model parameters. To further encourage the class-specific feature representation, we develop a Centroid-Enhanced (CEd) method to guide the update of these task-aware parameters. This approach is designed to adaptively minimize the distances between intra-class features while simultaneously maximizing the distances between inter-class features across all seen classes. The utility of this centroid-enhanced method extends to all "training from scratch" CIL algorithms. Extensive experiments were conducted on CIFAR-100 and ImageNet100 under different settings, which demonstrates that TaE achieves state-of-the-art performance.