The allen stud will only turn once you have loosened the large lock nut surrounding it, it should move freely then, however, I think you should be looking elsewhere before attempting this, the steering joints, bushes and UJs are far more likely to be the cause IMHO.
You can of course check the play in the box, by watching the input and output shaft whilst rocking the steering, but as above, get a mate to help you, and you will probably find the problem pretty quickly.
BTW, tyres shouldn't make the mechanics of the steering any different, only the 'feel' and vagueness whilst in a straight line and moving.