Possible Clues:

• Read quickly
• Like some healthy milk
• Glanced over
• Read superficially, Kipling tale is set in southern sea
• Went over cursorily
• Girl going into small expanse of water took top off
• Looked over quickly
• Read cursorily
• I'm grand in the broken desk from off the surface (7)
• Grazed the bar in a Milk Cup tie?