One can use one's

discretion
_NOUN

2 The freedom to decide what should be done in a particular situation.

He believes officers should use their discretion when dealing with speeding offences.

Rules say bus drivers can use their discretion regarding dogs but they must have a valid reason for refusing.

From Lexico.

  For Example 1:

The office typically requires employees to leave by 7:00 PM, but since the project deadline is approaching the manager uses their discretion to allow a few developers to stay late.

One can use one's

discretion
_NOUN

2 The freedom to decide what should be done in a particular situation.

He believes officers should use their discretion when dealing with speeding offences.

Rules say bus drivers can use their discretion regarding dogs but they must have a valid reason for refusing.

From Lexico. For Example 1:

The office typically requires employees to leave by 7:00 PM, but since the project deadline is approaching the manager uses their discretion to allow a few developers to stay late.

For a word beginning with the suggested L there could be

liberty
_NOUN

1.2 A right or privilege, especially a statutory one.

From Lexico. For Example 1:

The office typically requires employees to leave by 7:00 PM, but since the project deadline is approaching the manager has the liberty to allow a few developers to stay late.