He can speak standard English.
So I think that 一口 can be omitted and still the sentence can make sense.
How actually 一口 work in this sentence?
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口 is a classifier for some nouns that are related to the mouth, for example, speech, taste, and bite
北京腔 = Beijing accent --> 一口北京腔 = A mouthful of (a full language of) Beijing accent
大蒜味 = garlic taste/ smell --> 滿口大蒜味 = A mouthful of garlic taste/smell
吃鱼肉 = eat fish meat --> 吃几口鱼肉 = Eat a few bites of fish meat
他会说标准的英语 = he can speak standard English
他会说(一口)标准的英语 = he can speak a mouthful of (a full language of) standard English
If we change the classifier from 口 to 句 and write "他会说(一句)标准的英语", it would mean "he can speak (a sentence of) standard English" -- We don't know if he has a full vocabulary or not