As a scientist-turned-theologian, Polkinghorne refutes the idea of a "god of the gaps"; rejects the idea that God is God only of the unknown – which is a bit of a Cheshire Cat thing with God growing fainter and smaller as we discover more. Instead, Polkinghorne holds to a robust view of God as God of all things and all truth – thus, as we know more about nature and the physical world, we do not need God less or know less of him; rather, we see God's power as even greater – and we know him better.
On prayer: Polkinghorne talks about quantum unpredictability and the relation to a space for God to intervene and act. "Can a scientist pray?" Chaos theory teaches us that randomness occurs within an orderly structure – cloudy spaces exist where multiple possibilities exist and petitionary prayer may work. Reference Quarks, Chaos and Christianity – we can still believe in our action for the world and God's action, too.
20th century science has seen the death of a clock-work (wind up and let run) view of the world so that there is room for things to change in unexpected way. The world is not merely mechanical. We are not mechanical. Our prayer is a cooperation with God – not a mere magic or wish-granting.
The picture we now have of the world is much more complex than the traditional 18th or 19th century Modern view of the cosmos as a mere clockwork. There are clockworks within the cosmos, but there are also there pockets of randomness and cloudy areas where these is room to maneuver. It is in these cloudy spaces that prayer may be open. Most of human life is cloudy, too.
Further, these cloudy places will not be readily discerned as in saying that "God did this" or "God did that".
It should be noted that Polkinghorne doesn't hold to a strong view of God's foreknowledge (neither do I) and is probably rightly considered an open theist. He also has no problem seeing the machinations of the physical world including biological evolution as a part of God's glorious creation. God is not a puppet-master, but a conductor of a creative process. He sees God as reacting and improvising with creation, not having laid it out in advance and set it in stone - rigidly and inflexibly.
Polkinghorne holds that God created a world where things grow and change and happen - that is fruitful, certainly, but that it contained within it the possibility of suffering and evil. God is on the adventure with us - certainly outside it and greater than it is and with a viewpoint so comprehensive and overarching that God is not surprised in the sense that we are by what happens. But he is also not directing everything that occurs, and is adapting his action to the response of the creation.
He has created a world where dramatic possibilities existed within his creative framework - possibilities for both good and evil - and that this was the best answer that could be imagined. This is not the sentimental answer that our pious tradition would occur, but it seems to be a more honest answer.
Regions where real novelty occurs are always regions at the edge of chaos. Moving beyond this horizon, things are either locked into clockwork or fall apart in randomness.
Powerful and thoughtful stuff! Check it out!