ClearAll[show]

Framed@DynamicModule[{t = 0.001, i = 1}, s = 0;
  step = .05;
  trajnormal = 
   RandomFunction[PoissonProcess[1], {0, 100}]["PathFunction"];
  trajdiscount = 
   RandomFunction[PoissonProcess[5], {0, 100}]["PathFunction"];
  (*show[1] is trajectory of Poisson[1] process*)
  show[1, t_] := 
   Plot[50 - trajnormal[x], {x, 0.001, t}, PlotPoints -> 1000, 
    PlotRange -> {{0, 70}, {0, 55}}, ImageSize -> 500];
  (*show[2] is trajectory of Poisson[1] with Poisson[
  5] added at time of click=s*)
  show[2, t_] := 
   Plot[50 - trajnormal[x] - trajdiscount[Max[x - s, 0]], {x, 0, t}, 
    PlotRange -> {{0, 70}, {0, 55}}, ImageSize -> 500, 
    PlotStyle -> Red, PlotPoints -> 1000];
  EventHandler[
   EventHandler[
    Column[{Dynamic[i], 
      Dynamic[t = If[trajnormal[t] <= 50, t + step, t]], Dynamic[s], 
      Dynamic[show[i, t]]}], "MouseDown" :> (i = Mod[i, 2] + 1)], 
   "MouseDown" :> (s = t), PassEventsUp -> True]]