Alot of people who visit Copenhagen spend a day at Louisiana Museum of Modern Art in Humlebæk, about an hour north of the city. The museum (which is just beautiful and definitely worth a visit!) has a café that serves decent but slightly overpriced food. So here is my alternate suggestion: take the train just two stops further north and visit Gorm's.
Here you'll find a beautiful restaurant located just by the sea and a charming little harbour. Gorm makes some of the best pizzas in the country and the one with parma ham and thin slices of potato is amazing. They also serve Italian style 'smørrebrød' which is Danish for open faced sandwiches plus some delicious antipasti.
The food is great, the place is beautifully designed, prices are reasonable and the location couldn't be any better! During summer you can order a picnic basket with antipasta, bread, sweets and cutlery for two and bring it down to the beach. That is a perfect Sunday in the making! Don't forget to bring swim suits.