General:
* Pergola: A structure with a latticework roof, often used for vines.
* Trellis: A framework of latticework, often used to support climbing plants.
* Gazebo: A small, freestanding structure with a roof, typically used for relaxation or entertaining.
* Summerhouse: A small, detached building used for relaxation or entertaining, often with a garden setting.
More specific:
* Bower: A shady, leafy retreat.
* Veranda: A covered porch or balcony.
* Loggia: A covered, open-air space, often with a view.
* Pavilion: A freestanding structure with a roof, often used for a specific purpose, such as a bandstand.
* Shady nook: A small, secluded spot.
The best word to use will depend on the specific characteristics of the structure and the desired meaning.