There are a number of ways to test for the absence/presence of a spatial signal in a completely observed fine-resolution image. One of these is a powerful nonparametric procedure called enhanced false discovery rate (EFDR). A drawback of EFDR is that it requires the data to be defined on regular pixels in a rectangular spatial domain. Here, we develop an EFDR procedure for possibly incomplete data defined on irregular small areas. Motivated by statistical learning, we use conditional simulation (CS) to condition on the available data and simulate the full rectangular image at its finest resolution many times (M, say). EFDR is then applied to each of these simulations resulting in M estimates of the signal and M statistically dependent p-values. Averaging over these estimates yields a single, combined estimate of a possible signal, but inference is needed to determine whether there really is a signal present. We test the original null hypothesis of no signal by combining the (Formula presented.) -values into a single p-value using copulas and a composite likelihood. If the null hypothesis of no signal is rejected, we use the combined estimate. We call this new procedure EFDR-CS and, to demonstrate its effectiveness, we show results from a simulation study; an experiment where we introduce aggregation and incompleteness into temperature-change data in the Asia-Pacific; and an application to total-column carbon dioxide from satellite remote sensing data over a region of the Middle East, Afghanistan, and the western part of Pakistan. Supplementary materials for this article are available online.