I have a Sony DSC-V1 which I wouldn't say is similar, but in any auto focus mode, when you press the shutter button halfway it takes it approx 2.5 secs to auto focus and take the shot. If you press the shutter button directly after the first shot it takes a further 2.5 secs to auto focus again and take the shot.

When I switch it to manual mode, I can get 2 -3 shots every second although this does depend on aperture priority, shutter speed, and focus settings. This was one of the main reasons I chose this camera, it's balance between auto settings and allowing full manual operation, and compact size.

Im not sure if the DSC-P10 allows to operate in manual mode, but it is the auto focus which is delaying you from taking rapid shots.