For each new leave request, should find the user's manager, and populate the user id into the lookup on the leave request record.

Populate user field Test class

@isTest 
private class UpdateLeaveRequestTEST {
    static testMethod void validateUpdateLeaveRequest() {
       Leave_Request__c LR = new Leave_Request__c();
       System.debug('Manager before insert'+ LR.Manager__c);

       // Insert Time Sheet 
    
       insert LR;
    
       // Retrieve the new time sheet 
    
       LR = [SELECT Manager__c FROM Leave_Request__c WHERE Id =:LR.Id];
       System.debug('Manager after insert' + LR.Manager__c);

       // Test that the trigger correctly updated the manager 
    
       System.assertEquals('005U0000000VldF', LR.Manager__c);
    }
}

Trigger to populate user fields on record

trigger UpdateLeaveRequest on Leave_Request__c (before insert, before update ){
  /**
  1. For each Leave Request being inserted add User Id value in in Set of User Ids.
  2. Fetch all Users whose Id is in the Set.
  3. Add these fetched Users in a Map <User Id, User object>
  4. For each leave request being inserted get User from the map and update the field values  
  **/  
  
  //holds User Ids
  Set<Id> setUserIds = new Set<Id>();
  
  //holds a list of Users
  List<User> lstUser = new List<User>();
  
  //holds key value pairs of User Id and User Object
  Map<Id, User> mapUserObj = new Map<Id, User>();

  //holds User object
  User UserObj;
   
  //For each leave request being inserted add User Id value in in Set of User Ids.
  for(Leave_Request__c LRObj : Trigger.new){
    if(LRObj.OwnerId != null){
      setUserIds.add(LRObj.OwnerId);
    }
  }
  
  //Fetch all Users whose Id is in the Set.
  lstUser = [select Id, Name, ManagerId from User where Id in :setUserIds Limit 1000];
  if(lstUser.size() == 0){
    return;  
  }
  
  //Add these fetched Users in a Map <User Id, User object>
  for(User usrObj : lstUser){
    mapUserObj.put(usrObj.Id, usrObj);  
  }
  
  //for each leave request being inserted get User from the map and update the field values
  for(Leave_Request__c LRObj : Trigger.new){
    //get User object
    if(LRObj.OwnerId != null){
      if(mapUserObj.containsKey(LRObj.OwnerId)){
        UserObj = mapUserObj.get(LRObj.OwnerId);
        //map leave request fields to User fields
        LRObj.Manager__c = UserObj.ManagerId;
        LRObj.Owner2__c = UserObj.Id;
      }  
    }
  }
}