As dependence on Content Delivery Networks (CDNs) increases, there is a growing need for innovative solutions to optimize cache performance amid increasing traffic and complicated cache-sharing workloads. Allocating exclusive resources to applications in CDNs boosts the overall cache hit ratio (OHR), enhancing efficiency. However, the traditional method of creating the miss ratio curve (MRC) is unsuitable for CDNs due to the diverse sizes of items and the vast number of applications, leading to high computational overhead and performance inconsistency. To tackle this issue, we propose a lightweight and adaptive cache partitioning scheme called $\varepsilon$-LAP. This scheme uses a corresponding shadow cache for each partition and sorts them based on the average hit numbers on the granularity unit in the shadow caches. During partition resizing, $\varepsilon$-LAP transfers storage capacity, measured in units of granularity, from the $(N-k+1)$-th ($k\leq \frac{N}{2}$) partition to the $k$-th partition. A learning threshold parameter, i.e., $\varepsilon$, is also introduced to prudently determine when to resize partitions, improving caching efficiency. This can eliminate about 96.8% of unnecessary partition resizing without compromising performance. $\varepsilon$-LAP, when deployed in PicCloud at Tencent, improved OHR by 9.34% and reduced the average user access latency by 12.5 ms. Experimental results show that $\varepsilon$-LAP outperforms other cache partitioning schemes in terms of both OHR and access latency, and it effectively adapts to workload variations.