Try this. I prefer to use a for loop (rather than a for each loop) here.
Code:
Sub Rpl()
Dim lineText As String
Dim MyPos
Dim x As Integer
For x = 1 To ActiveDocument.Paragraphs.Count
lineText = ActiveDocument.Paragraphs(x).Range.Text
MyPos = InStr(lineText, "Test Text")
If MyPos = 1 Then
ActiveDocument.Paragraphs(x).Range.Text = lineText & "LxW "
End If
Next x
End Sub
Also, notice that in your original you only change the string -- lineText. The actual paragraph -- singleLine -- is never changed.