Looks like Davao City's only international flight is SilkAir from Singapore .. so the only one stop option would be via Singapore connecting onto Silkair

That's likely to be the simplest and smoothest option (though probably not the cheapest)

Otherwise you'd likely have to go via Manila and there's no direct flights from the UK to Manila so you'd be having to connect somewhere and then connect at Manila (which is a bit of a crappy airport to connect at )

Domestic flights are very cheap in the Phillipines with several competing carriers offering super low fares (Cebu Pacific is probably the best of the low cost guys .. also there's Zest Air and Air Phil Express who have cheap fares)