There are several restaurants on what's called Cleveland Place in Manhattan that have outdoor "gardens" in the back where you can eat. Cleveland Place is actually the east side of Lafayette Street just below Spring Street. The one I know and recommend used to be called Tratoria Vanessa & Pietro, but has since changed management and names. But it's the only Italian restaurant on the strip, and the food is quite good for what's basically a neighborhood Italian joint. Also, they've decorated the space, so it's quite attractive. These places are always crowded in the summer, so if you want to eat there, you have to make a reservation or be prepared to wait a while.
The great thing about eating out back is that you avoid the unpleasantness of eating on the street.
Another popular outdoor garden out back is the Miracle Grill in the East Village.