Your example runs fine on my system. Windows 10 Home.

I will say that I get more smooth display with the following:

Manipulate[
  DynamicModule[{gr},
  Dynamic[
   If[run, \[Phi] += d\[Phi]]; x = Cos[\[Phi]]; y = Sin[\[Phi]]; 
   gr = Graphics[{Red, Point[{x, y}], Blue, Point[{0., 0.}], 
      Opacity[0.25, Gray], Rectangle[{-1.1, -1.1}, {1.1, 1.1}]}, 
     PlotRange -> 1];
   Show[gr]
   ]
  ], {run, {False, True}},
 {{d\[Phi], 0.1}, 0., 0.2},
 {x, ControlType -> None}, 
 {y, ControlType -> None},
 {{\[Phi], 0.}, ControlType -> None}
 ]

Ulrich,

I think your use of SynchronousUpdating->True is a more desirable solution and it makes more sense to me from the documentation (ie, the Frontend blocks other tasks during evaluation when set to True) . Using Dynamic has a similar effect, but the reason appears to be more subtle. But based on its documentation, I think the use of Dynamic forces the expression within Manipulate to evaluate whenever any dependent variables change values, not just when a control variable changes. I'm not an expert on Dynamic, however.

Dear Henrik, please tell me whether on your system the cell braked right to the running graphics shows unsteady flickering as it does on mine? This flickering seems to result from busy communication between front end and kernel and could be an indicate the problem that the program is causing on my system.

Thanks for your help.

Dear Henrik, do you understand you understand the role of this additional Dynamic? The Documentation says a bit about Dynamic within Manipulate and explains that parts of the first argument of Manipulate that are not included by the additional Dynamic are not automatically revaluated. But if everything is within the additional Dynamic there should be the same behavior as without the additional Dynamic. Seems to be a difficult matter on which the Documentation could be a bit more explicit. As I mention in my reply to David, a final addition SynchronousUpdating ->True has the same beneficial effect as the additional Dynamic.

Best Regards, Ulrich

In this case, Animate might be a better choice than Manipulate

Animate[
    Graphics[{
        PointSize@Large, Red, Point[{Cos@a, Sin@a}],
        Blue, Point[{0, 0}],
        Opacity[0.25, Gray], Rectangle[{-1.1, -1.1}, {1.1, 1.1}]
    }],
    {a, 0, Infinity, 0.1},
    AnimationRunning -> False
 ]