Claudius is generally really good at politics – but this sending Hamlet to England business is kind of a political mistake. With Hamlet gone, people are free to think whatever they want and they are likely to be predisposed to be on Hamlet’s side. So by sending Hamlet away, he’s got no murderer and thereby probably casts some suspicions on himself. If he’d kept Hamlet nearby, he’d have had to have some kind of public justice. Hamlet would likely not deny his murder of Polonius. I guess the danger, thought, would be that in confessing to Polonius’ murder, he might reveal why he killed him, which might reveal Claudius’ actual murder.