A good place for something more substantial than your standard coffee shop stuff, but still the same kind of atmosphere, is a little cafe type place on that road where the Golden Cross/Newt & Cucumber are (Bayley Lane?)
If you turn down from the courtyard end it's about halfway down on the left. I went there on a Saturday morning before a city game, had a perfectly reasonable omelette and chips, and the lady I was with had vegetarian breakfast, but there was loads of choice, along with coffees/teas/etc. It may well possibly have been licensed as well, although it was only around 10am so I didn't really pay much notice to that....
Also around there, just about opposite the Courtyard, on the High Street, is a little Chinese cafe/restaurant. They do some awesome stuff, very no-frills, but good food and portions.
Edit: it's on Hay Lane, not Bayley Lane, sorry, and it's called Playwrights. The Chinese place is called Chi Bar.