Mappings play an important role in environmental science applications by allowing practitioners to monitor changes at national and global scales. Over the last decade, it has become increasingly popular to use satellite imagery data and machine learning techniques (MLTs) to construct such maps. Given the black-box nature of many of these MLTs though, quantifying uncertainty in these maps often relies on sampling reference data under stricter conditions. However, practical constraints can sampling such data expensive, which forces stakeholders to make a trade-off between the degree of uncertainty in predictions and the costs of collecting appropriately sampled reference data. Furthermore, quantifying any trade-off is often difficult, as it will depend on many interdependent factors that cannot be fully understood until more data is collected. This paper investigates how a combination of Bayesian inference and an adaptive approach to sampling reference data can offer a generalizable way of managing such trade-offs. The approach is illustrated and evaluated using a woodland mapping of England as a case study in which reference data is collected under constraints motivated by COVID-19 travel restrictions. The key findings of this paper are as follows: (a) an adaptive approach to sampling reference data allows an informed approach when quantifying this trade-off; and (b) Bayesian inference is naturally suited to adaptive sampling and can make use of Monte Carlo methods when dealing with more advanced problems and analytical techniques.