How can I eliminate play in a water tap?
From your comment, it sounds like the root issue is the lower water pressure from your hot supply.  With lower water pressure, when you only slightly open the valve, there might not be enough pressure and/or flow for the water to overcome the other forces acting against it (gravity, restriction from valve, etc.).  Similar to say a garden hose - with low pressure, it might not leak at all, but open the valve up all the way and it starts leaking at certain places (except in this case, the leak is the flow you want).

The first thing to check is that the shutoff for your hot water supply is fully open - at the sink, at the tank and anywhere else along the way where there might be a shutoff. If all of the shutoffs are fully open then it could be due to incorrectly sized pipes, mineral build up (depends on type of pipe).  Check to see if the pressure is low at any other facuets - this will give you a good idea if its a problem with the plumbing or just that particular fixture.