A separate construction like that is called "cenador". Depending on the region and specifications I'm not quite familiar with (materials, design, etc.), it may be also called "pérgola", "kiosko" (or "quiosco"), "templete", "glorieta", "emparrado" (this last one if it has plants growing on it)...
It may also be a "cobertizo", as Poli said, but I associate that term with a roof that comes as an extension of the house, rather than an autonomous construction.
As for "casa de almacenaje", this is only true if you use it to store things, but this seems to be used more to spend a comfortable afternoon/evening in there.
__________________
♪ ♫ ♪ Ain't it wonderful to be alive when the Rock'n'Roll plays... ♪ ♫ ♪
|