How do I deal with hydrophobic soils ?

Try mixing wood chips into the soil, anything that will absorb the water, but not thorn or pine. It's a bit more difficult with the lawn but you could spike it and drag out anything that isn't grass.

Thanks for your suggestion Clover, I'll give it a try.

Spiking is essential - probably quite regularly till it gets better. Is the ground very compacted - like a clay-type soil? If so you need to add oodles of compost to help break it down and allow the water in. Sand (sharp) would probably be helpful too. Also could it be that trees surrounding your garden are taking the moisture? Does the water literally run off, or does it just pass through quickly and leave the soil still dry?