Few-shot action recognition (FSAR) has recently made notable progress through set matching and efficient adaptation of large-scale pre-trained models. However, two key limitations persist. First, existing set matching metrics typically rely on cosine similarity to measure inter-frame linear dependencies and then perform matching with only instance-level information, thus failing to capture more complex patterns such as nonlinear relationships and overlooking task-specific cues. Second, for efficient adaptation of CLIP to FSAR, recent work performing fine-tuning via skip-fusion layers (which we refer to as side layers) has significantly reduced memory cost. However, the newly introduced side layers are often difficult to optimize under limited data conditions. To address these limitations, we propose TS-FSAR, a framework comprising three components: (1) a visual Ladder Side Network (LSN) for efficient CLIP fine-tuning; (2) a metric called Task-Specific Distance Correlation Matching (TS-DCM), which uses $\alpha$-distance correlation to model both linear and nonlinear inter-frame dependencies and leverages a task prototype to enable task-specific matching; and (3) a Guiding LSN with Adapted CLIP (GLAC) module, which regularizes LSN using the adapted frozen CLIP to improve training for better $\alpha$-distance correlation estimation under limited supervision. Extensive experiments on five widely-used benchmarks demonstrate that our TS-FSAR yields superior performance compared to prior state-of-the-arts.