We consider the detection of a correlated random process immersed in noise in a wireless sensor network. Each node has an individual energy constraint and the communication with the processing central units are affected by the path loss propagation effect. Guided by energy efficiency concerns, we consider the partition of the whole network into clusters, each one with a coordination node or \emph{cluster head}. Thus, the nodes transmit their measurements to the corresponding cluster heads, which after some processing, communicate a summary of the received information to the fusion center, which takes the final decision about the state of the nature. As the network has a fixed size, communication within smaller clusters will be less affected by the path loss effect, reducing energy consumption in the information exchange process between nodes and cluster heads. However, this limits the capability of the network of beneficially exploiting the spatial correlation of the process, specially when the spatial correlation coherence of the process is of the same scale as the clusters size. Therefore, a trade-off is established between the energy efficiency and the beneficial use of spatial correlation. The study of this trade-off is the main goal of this paper. We derive tight approximations of the false alarm and miss-detection error probabilities under the Neyman-Pearson framework for the above scenario. We also consider the application of these results to a particular network and correlation model obtaining closed form expressions. Finally, we validate the results for more general network and correlation models through numerical simulations.