Adapting a single large language model (LLM) to perform reliably across diverse and sometimes conflicting domains remains a practical challenge. Existing fine-tuning approaches each carry notable limitations: full fine-tuning on mixed datasets is susceptible to knowledge interference and catastrophic forgetting; parameter-efficient tuning (PEFT) reduces cost and mitigates forgetting but often lacks sufficient architectural capacity for multi-task specialization; and traditional Mixture-of-Experts (MoE) models typically require expensive pre-training from scratch. Motivated by these gaps, we propose LMoE-FAAT, a framework that constructs a Lightweight Mixture-of-Experts using LoRA-based parameters and incorporates a future-aware auxiliary loss mechanism based on multi-token prediction. Specifically, we convert a pretrained dense LLM into a Lightweight Mixture-of-Experts (LMoE) by inserting LoRA-based experts into attention and feed-forward layers, enabling more modular allocation of domain-specific knowledge with minimal parameter overhead. In addition, we incorporate Future-Aware Auxiliary Tuning (FAAT), a multi-token prediction objective intended to enrich the training signal without increasing inference-time cost. Preliminary experiments on a multi-task benchmark indicate that LMoE-FAAT can provide consistent improvements over full fine-tuning and several widely used PEFT methods, particularly in the context of Qwen models. These findings suggest that combining lightweight expert modularity with a future-aware auxiliary objective may offer a promising direction for more efficient and scalable multi-task LLM adaptation. Our repository is open-sourced via https://github.com/huang-ml/LMOE-FAAT .