ClearAll[myPrint];
myPrint[exp_] := Module[{dir, fns, nb},
  dir = NotebookDirectory[];
  fns = FileNames[FileNameJoin[{dir, "myMessagesTemp.nb"}]];

  If[fns === {},
   nb = CreateNotebook[];
   NotebookSave[nb, FileNameJoin[{dir, "myMessagesTemp.nb"}]],
   nb = NotebookOpen[fns[[1]]]
   ];

  NotebookWrite[nb, Cell[exp, "Output"]]
  ]

Module[{},
 myPrint["Hello World"];
 myPrint["Hello Again"];
 Do[myPrint[n], {n, 4}]
  Do[myPrint[ToString[n]], {n, 4}]
  myPrint[x^2];
 ]