Hi Clare. Thanks for writing MyGolfInstructor.com.
When golfers hit their puts "fat" it's usually because of one of three reasons.
1) They have too much tension in their hands and arms which forces them to push the putter downward.
2) They are excessively overactive with their hands and arms. For instance their arms can start sligtlty bent, but then they either straighten them out or move them away from their body.
3) They have too much body movement. For instance dropping the head or spine in the stroke can make the putter hit the ground.
The first thing I would recommend is that you do hover the club just a little and try a very light pressure. You may even try the Pincher Grip found here:
It will help you take any tension or desire to force it out of your stroke.
The next thing I would recommend is practicing with your head against a wall to make sure you are not swaying it back and forth or moving it up and down. This drill will really help you to stabilize. You can find that drill called the Baseboard Drill here:
Give those two things a try and let me know how it goes!
By: Maria Palozola