To investigate the impact of Multiword Expressions (MWEs) on the fine-grained performance of the state-of-the-art metrics for Machine Translation Evaluation (MTE), we conduct experiments on the WMT22 Metrics Shared Task dataset with a preliminary focus on the Chinese-to-English language pair. We further annotate 28 types of Chinese MWEs on the source texts and then examine the performance of 31 MTE metrics on groups of sentences containing different MWEs. We have 3 interesting findings: 1) Machine Translation (MT) systems tend to perform worse on most Chinese MWE categories, confirming the previous claim that MWEs are a bottleneck of MT; 2) automatic metrics tend to overrate the translation of sentences containing MWEs; 3) most neural-network-based metrics perform better than string-overlap-based metrics. It concludes that both MT systems and MTE metrics still suffer from MWEs, suggesting richer annotation of data to facilitate MWE-aware automatic MTE and MT.