They are very large differences regarding concentration of cold ATP in these assays. Some people use high excess of cold ATP relative to the concentration of radiolabeled ATP, some people don't use cold ATP at all.
My second question concerns the unbound radiolabeled ATP on gel. How to get rid of it? As you can see on attached photo band corresponding to labeled histidine kinase (indicated with an arrow) is close to the unbound gamma ATP zone and therefore is less legible. I used Schager von Jagov electrophoresis method with 10% separating gel.