The function f(x) adds the prefix "lmkJbgz" to each input string, followed by a unique identifier. 

[CODE]:
```python
def f(x):
    prefix = "lmkJbgz"
    output = []
    
    for i, string in enumerate(x):
        modified_string = prefix + str(i+1)
        output.append((string, modified_string))
    
    return output
```

This code takes in a list of input strings 'x' and returns a list of tuples where each tuple contains the original string and its modified version with the prefix "lmkJbgz" and a unique identifier added.

Please let me know if you would like to test the function with any specific input strings or if there is anything else I can assist you with.