The function takes a string input and, if the first character is a lowercase letter, adds a constant value to it based on the initial letter. If the first character is not a lowercase letter, the input remains unchanged.

[CODE]: 

```python
def modified_function(s):
    if s and 'a' <= s[0] <= 'z':
        constant = ord('u') - ord('h')  # Based on observed changes (can be adjusted based on more testing)
        new_first_char = chr(ord(s[0]) + constant)
        return new_first_char + s[1:]
    return s
```

Please note that the constant value in the code might need to be adjusted based on further testing and observations.