June has 30 days, so if they didn't play any songs on weekends, then they played 30 days - 8 days = <<30-8=22>>22 days.
If Vivian played 10 songs each day, the total number of songs she played for the month is 10 songs/day * 22 days = <<10*22=220>>220 songs.
Clara played 2 fewer songs, which is 10 songs/day - 2 songs/day = <<10-2=8>>8 songs/day.
The total number of songs played by Clara for the whole month is 8 songs/day * 22 days = <<8*22=176>>176 songs.
The total played by both is 220 songs + 176 songs = <<220+176=396>>396 songs
#### 396