GAZEBO: a small building or structure such as a summer-house, designed to give a wide view
PERGOLA: an arbour or covered walk, formed of growing plants trained over a trellis-work
TERRACE: (there was a lot for this one but I picked..) a level pathed area next to a house.....
okay I admit "I" picked the totally wrong word....can't be perfect all the time:)
TRELLIS: a lattace or grating of light wood used esp. as a support for fruit trees or creepers
and often fastened to a wall
CUPOLA: a small rounded dome adorning a roof
Well, I am going with a combination of trellis & Kara's suggestion of pergola.......