That play you are talking about is known as "Backlash" and yes there are a few ways to combat that. First is software compensation. You take the measurments of backlash in your machine. With a dial indacator. and then program it into your controller software. What it will do is move to the desired point, then reverse the amount specified, to eliminate the backlash. There are also Special nuts you can get or make, if I rmember right they are called BSA nuts. These will tigthen up the machines lead screws. You will have to do more research on this.
I have the desktop CNC mill "MAXNC 15 Open loop" See here
www.maxnc.com You will need motors, power supply and controller, there are in fact many software packages on the market (Mastercam 5 comes to mind) that are programable to your setups specs.
You will also need to dowhat morph has done and design some motor mounting plates
I wish ya luck, as with anything its a pain in the a$$ to do, but well worth it in the end.
My english is fine, butmy spelling/typing sucks! So dont feel bad.