I am travelling to London with a group of youngsters in August and we would like to go to an open-air cinema or concert in the eve. We are looking for some free events between 5 - 10th of August.
There are a bunch of free summer concerts and plays at The Scoop at More London which is near London Bridge, by the Mayor's Office. You can google for schedules but, just as an example, the Free Open Air Theatre will be holding evening plays in early August (note that you should arrive early and consider bringing a picnic but no glass allowed).
St Martin in the Fields runs a series of classical music concerts (including the very popular free lunchtime ones) held in the church or on the grand stairs in front of Trafalgar Square. You can check the schedule and ticketing. Note that in London many FREE events are ticketed due to crowd control factors. Other churches that hold regular free lunchtime recitals include St Olav and St James Piccadily.
Kings Place has free jazz concerts on Friday nights in August, although I think they may be in the bar area (?) so may be less appropriate for kids. The Notting Hill Arts Club holds cool Saturday night concerts and you get in for free if you arrive before 9pm (is that too late?). The Royal Albert Hall will be holding its Proms series and you can get on the day (line up early) gallery tickets for cheap (from 5GBP) -- concerts include a variety of classical music, jazz, indie and film related, the venue is stunning!
For film, it is a bit harder to find free screenings unless you search for specific days (you might get lucky). You can however find cheap options (5-15GBP) for open air screenings with The Floating Cinema, The Luna Cinema, Nomad Cinema, etc.
You might want to use Visit London as a resource and email ther events team directly for date-specific recommendations.
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Time Out is your best guide; http://www.timeout.com/london/film/outdoor-cinema-in-london