Proliferation of cloud-based latency-sensitive workloads requires infrastructures tuned to their workload-specific latency constraints. Today, they shape the cloud from a generalized computing platform to diverse workload-specific cloud environments. As the demand for latency-sensitive workloads increases, cloud service providers continue to scale their infrastructure, adversely increasing the carbon footprint and challenging climate-crisis-driven net-zero emission goals. Due to performance-oriented rigid deployment patterns of latency-optimizations, reducing its carbon footprint is challenging. Therefore, efficient techniques that exploit application specific opportunities are needed in that. To this end, we present a detailed taxonomy of recent literature on carbon-aware resource management in latency-sensitive cloud computing environments. Using the taxonomy, we analyze existing works discussing their optimization aspects, identify the gaps, and highlight future research directions.