It's an interesting scenario, the more I think about it. I assume that there would be some level of German-American backlash against the party in power, although it might be more muted due to existing GOP strength in the Midwest at the time. If I recall correctly, there was a small depression around 1920, which would have likely hurt the GOP. Additionally, I'm not sure how the First Red Scare plays out under a Hughes administration, assuming that swings the electorate much at all (my best guess would be 'maybe', considering there was an element of nativist bigotry in the backlash against perceived political radicals, like the kind later seen in the Sacco and Vanzetti trial, that might affect some ethnic groups' voting behavior).
I think Hughes still wins, but the Midwest could be a wild card, especially if Democrats still nominate James M. Cox. Eugene Debs probably still takes 3% or so of the popular vote. A lot of variables involved, but the end result is significantly closer than Harding's landslide IOTL. I'd guess maybe a 3-6% PV win for Hughes overall?
I agree, I also think that prohibition would be a factor. In our world it was blamed equally on both parties but in this the GOP would be solely to blame and this would probably lead to Cox holding onto the big cities which means a narrower margin in Massachusetts and New York and possibly his outright flipping Maryland.