The authors investigate how to detect different tail behaviors of two risk random variables with the same mean. A class of convex risk measures, referred to as adjusted higher-order expected shortfall (ES), is introduced and studied. The adjusted risk measure quantifies risk as the minimum amount of capital that has to be raised and injected into a financial position to ensure that its higher-order ES does not exceed a pre-specified threshold for every probability level. This new risk measure is linked to dual higher-order increasing convex order by choosing the risk threshold to be the higher-order ES of a special benchmark random loss. The dual representation for adjusted higher-order expected shortfall is also given.