We consider cache-aided wireless communication scenarios where each user requests both a file from an a-priori generated cacheable library (referred to as 'content'), and an uncacheable 'non-content' message generated at the start of the communication session. This scenario is easily found in real-world wireless networks, where the two types of traffic coexist and share limited radio resources. We focus our investigation on single-transmitter wireless networks with cache-aided receivers, where the wireless channel is modelled by a degraded Gaussian broadcast channel (GBC). For this setting, we study the (normalized) delay-rate trade-off, which characterizes the content delivery time and non-content communication rates that can be achieved simultaneously. We propose a scheme based on the separation principle, which isolates the coded caching problem from the physical layer transmission problem, and prove its information-theoretic order optimality up to a multiplicative factor of 2.01. A key insight emerging from our scheme is that substantial amounts of non-content traffic can be communicated while maintaining the minimum content delivery time, achieved in the absence of non-content messages; compliments of 'topological holes' arising from asymmetries in wireless channel gains.