I'm starting to think I must know a different Cambridge from everyone else.....
Magpie, there used to be several Tatties in town, but afaik, there's only one branch left now. Yummy spuds though. Also, the only Carringtons Cafe I know in town is nowhere near the Grafton - it's on Market Street, opposite TKMaxx. There might be another branch by the Grafton, though - I never go to that side of town.
Boleh, there isn't an HSBC in the Market Square! The only HSBC in the town centre is on St Andrews Street. I think there used to be one near the market several years ago, but definitely not for the last few years. There is a cafe called Indigo, though, next to the entrance to the Arts Theatre. (OP, it's down an alley way called "St Edwards Passage", if you can't find it ask a local for the Arts Theatre).
There are two cafes on the Market Square with outside seating. One is Italian, called Don Pasquales, the other is called Stazione. They are both more posh and expensive than your average cafe.
There's a Bill's in Green Street, which is my preference for lunch. But if you want to be outside people watching, then it's not the place to be. Great ambience inside though.
If you don't mind spending a little bit more, there's a restaurant called "Galleria" overlooking the river on Bridge Street. However, you can get the same vibe much more cheaply from buying an ice cream by Scudamore's Punting Co and sitting by the river!
The Tourist Information Office, opposite the Cath Kidston shop, has a cafe with outside seating. It's just off the Market Square. I have never eaten there, though, so can't comment on the food.
I'm sure you'll find somewhere nice, have a lovely day with your daughter.