Works fine:
ContourIntegrate[1/(z - a), z \[Element] RegionBoundary[Disk[{0, 0}, 1, {Pi/6, 11 Pi/6}]]]
(*ConditionalExpression[Piecewise[{{(2*I)*Pi, (Abs[a] < 1 && Arg[a] > Pi/6) || (Abs[a] < 1 && Arg[a] < -1/6*Pi)}}, 0],
(Abs[a] != 1 || (Pi + 6*Arg[a] >= 0 && 6*Arg[a] <= Pi)) && ((6*Arg[a] != Pi && Pi + 6*Arg[a] != 0) || Abs[a] > 1) && a != 0]*)
Regards.