You make assumptions about how the file is read and where the write seek position is. Mode "r+" supports write-after-seek, however, you have to explicitly seek to that position.

with open('test.txt','r+') as f:
  f.seek(f.tell())
  f.write("wow!")
  f.flush()

test.txt now contains:

Helwow!orld!

See also Difference between modes a, a+, w, w+, and r+ in built-in open function?