There's little choice, but the food and drink is adequate for the location!
There is a perfectly good pub, serving great hot meals and the usual full range of alcoholic and soft drinks. The Haven House Inn has extensive tables out the front during the summer, which is where most people choose to sit and eat or drink. Inside the pub is quite small, with some tables (which can be reserved) and out the rear there's a small "smoking" terrace with 4-5 tables and a restricted water view.
Inside the pub has a "No Children" policy, but dogs are allowed. Children are not allowed inside at all, but are welcome outside. I suspect this is more of a space issue than a case of "we hate children"; the pub is so small inside there's simply not the space.
If you're into your history, then the car park at Mudeford Quay is the site of a battle in 1784, fought between pirates and customs men. The Haven Inn and the pub up the road called the Ship in Distress were both owned by the same landlady and she was the centre of a successful smuggling gang. George Coombes was hanged from a gibbet outside the pub - I'll leave it to you to do your research on this battle and George's execution, it'll give you something to use your Kindle for while you're sitting enjoying the great outdoors and a pint.
There's a substantial, high quality, fresh seafood stall, selling every variety of fish and seafood that you can think of that are caught locally. Everything in the seafood shop is freshly caught. Here you'll find small nibbles, such as whelks and cockles, as well as mussels, fresh crab and fresh lobster. As everything is freshly caught, it's hard to say what'll be on offer from day to day, but catch of the day can include Dover sole, sea bass, scallops, prawns. All the fish is always of the highest quality.
If you watched the television programme, Invasion of the Job Snatchers, aired in the UK in 2014, one of the participants, Amber Wood from Slough, had a job at the fish stall in Mudeford. In this TV programme unemployed young people from around the country were brought to Christchurch to try to get jobs and the TV show followed their trials and tribulations.
If you want more food than this, then Avon Beach is just 1/4 of a mile away by walking along the beach path, which is wide, concrete and flat, so again, very accessible-friendly.
Mudeford Quay is the sort of place you might go to for a meal and a drink and to then collect some fresh fish to take home and leave.