One can use one's

> **discretion**  
> <sub>NOUN</sub>  
>  
> **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](https://www.lexico.com/definition/discretion). 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.

<HR>

For a word beginning with the suggested **L** there could be

> **liberty**  
> <sub>NOUN</sub>  
> 
> **1.2** A right or privilege, especially a statutory one.


From [Lexico](https://www.lexico.com/definition/liberty). 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.