I have the Sony H5 and also was disappointed with the action shots but then I discovered a way to make it somewhat better!
I set it of course to the action setting (little man golfing) but also I set it to BRK which is on (the H5 at least) in between the shutter button and the setting dial, it has Normal, Multi burst and Bracket and it's the Bracket that seems to work best.
It takes a series of 3 shots in a row and works best outside in the daylight.
You may want to give that a try and see if it works for you, I mean obviously it's not as good as a DSLR but the smaller size and comfort factor are important too, I find that other than the action shots over all the camera is pretty good!
Here is a before
and an After with the BRK