Speech emotion recognition (SER) plays a critical role in building emotion-aware speech systems, but its performance degrades significantly under noisy conditions. Although speech enhancement (SE) can improve robustness, it often introduces artifacts that obscure emotional cues and adds computational overhead to the pipeline. Multi-task learning (MTL) offers an alternative by jointly optimizing SE and SER tasks. However, conventional shared-backbone models frequently suffer from gradient interference and representational conflicts between tasks. To address these challenges, we propose the Sparse Mixture-of-Experts Representation Integration Technique (Sparse MERIT), a flexible MTL framework that applies frame-wise expert routing over self-supervised speech representations. Sparse MERIT incorporates task-specific gating networks that dynamically select from a shared pool of experts for each frame, enabling parameter-efficient and task-adaptive representation learning. Experiments on the MSP-Podcast corpus show that Sparse MERIT consistently outperforms baseline models on both SER and SE tasks. Under the most challenging condition of -5 dB signal-to-noise ratio (SNR), Sparse MERIT improves SER F1-macro by an average of 12.0% over a baseline relying on a SE pre-processing strategy, and by 3.4% over a naive MTL baseline, with statistical significance on unseen noise conditions. For SE, Sparse MERIT improves segmental SNR (SSNR) by 28.2% over the SE pre-processing baseline and by 20.0% over the naive MTL baseline. These results demonstrate that Sparse MERIT provides robust and generalizable performance for both emotion recognition and enhancement tasks in noisy environments.