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COVID-19 has a spectrum of disease severity, ranging from asymptomatic to requiring hospitalization. Understanding the mechanisms driving disease severity is crucial for developing effective treatments and reducing mortality rates. One way to gain such understanding is using a multi-class classification framework, in which patients' biological features are used to predict patients' severity classes. In this severity classification problem, it is beneficial to prioritize the identification of more severe classes and control the "under-classification" errors, in which patients are misclassified into less severe categories. The Neyman-Pearson (NP) classification paradigm has been developed to prioritize the designated type of error. However, current NP procedures are either for binary classification or do not provide high probability controls on the prioritized errors in multi-class classification. Here, we propose a hierarchical NP (H-NP) framework and an umbrella algorithm that generally adapts to popular classification methods and controls the under-classification errors with high probability. On an integrated collection of single-cell RNA-seq (scRNA-seq) datasets for 864 patients, we explore ways of featurization and demonstrate the efficacy of the H-NP algorithm in controlling the under-classification errors regardless of featurization. Beyond COVID-19 severity classification, the H-NP algorithm generally applies to multi-class classification problems, where classes have a priority order.