The function takes a string input and inserts an 'H' after the second character while incrementing the ASCII value of the third character by 2.

[CODE]: 
```python
def approx_f(x):
    return x[:2] + 'H' + chr(ord(x[2]) + 2) + x[3:]
```

You can further test the function, but from the tested inputs, this is the function's characteristic behavior.

If you would like me to test more inputs, please let me know.